Q:

Can a GM prohibit players from using external reference materials (like the PHB) during play?

I have recently started playing D&D 5th edition. All of the players are new except for the GM, who has played before, but it is his first time being a GM.
I am playing a Wizard character, and I have been keeping my spells for my spellbook in a note on my phone for easy reference during a battle. Now my GM is requiring that all members know the effects of their spells or risk having them fail. Is that even allowed?  Shouldn't I be able to reference my spells or I am just crazy?

After asking my GM about this, he has said he wants to stay true to the rules of the game and to make sure everything goes as it should, which I find reasonable. While I have been making sure to do research and to know what my spells are, in general, my other teammates haven't done the same research. They still try to use their spells and abilities without knowing how to use them. In that sense, I understand his rule, but for someone who has done the research, it seems adversarial as you said. There is no normal time limit - we just play.
His reasoning is because the character gains a new ability, the wizard gets two new spells every level, that doesn't mean they can use them 100% the first couple times, which I understand. Yet, he says for the spells that my character has "memorized", and uses well, that I should also have them memorized (casting time, effects, everything) without the use of a reference, a written notebook, or a spell sheet. He then went on to say that I could look at the spells, but that would count as my character's action. I am fairly sure that is not how the game is to be played.

A:

Can the GM ban references?
Yes.  Specifically, "the D&D rules help you and the other players have a good time, but the rules aren't in charge. You're the DM, and you are in charge of the game." (DMG p.4, "The Dungeon Master," emphasis in original.)
That being said...
This is strange, as presented. In nigh-thirty years playing I've not run across a GM who disallowed looking up references. Not during your turn? Sure. Not at all? Weird.
(When running a table that operates under time constraints I will often skip a player who's not "ready" for their turn in combat. This tends to fall harder on spellcasters than on martial types, and harder on new players than on experienced hands. Of course, this is articulated at the beginning as a standard of play.)
But according to your edit, that's not what's going on here.
Your GM seems to be basing this restriction on two ideas: the character needs some time to be able to "fully" use their class features, and the player's skill should impact the character's abilities.

The character can't use their class features--new spells, in this case--until they're "broken in."
This has no basis in the rules. In short, "Beyond 1st level" (PHB p.15) tells us that when your character earns certain XP they gain a level. The class descriptions tell us what new things the character can do when they gain a level.
@Kryan's answer has the right of this: the two new spells your wizard now knows by dint of increasing a level represent the work that character's already done to learn new spells, not some new task the character needs to take on.
(As an aside, do fighters only get part of their Ability Score Improvement or feat until they've sufficiently proven themselves at their new level? Do druids pop out of Wild Shape suddenly because they're not yet well-trained in a new form?)
Where there is some support for something like this is in the question of when a character earns XP or can gain a level. Adventurer's League rules, for instance, only allow a character to gain a level when they've completed a long rest or at the end of a module. That is, if killing goblin three of seven in an encounter would put you at a new level, we usually don't stop combat to do it then. 
This is also subtly achieved by GMs who award XP at the end of sessions, or by "milestoning." 
Again, though, once you've got the level, you get all the class features that come with it, full stop.
Player skill = character ability. Charging your character an action for you to look at your character's spells is... insane. That's a huge hit in the action economy.
This seems to stem from an idea that the player's ability to memorize everything redounds to the character's ability to perform in the fiction.
The idea that player skill should be important is an old one, and has plenty of merit to it, I think.
But this is an incredibly ham-handed way to bring player skill into the game.
(Again, I've got to wonder if your GM tries to stab the fighter-player just to see how well the player reflects DEX 18?)

"Shouldn't I be allowed to reference my spells...?"
This is hard to answer: there may be things going on that we're not aware of. What you should be able to do, without question, is talk to your GM about what's going on. Ask them what purpose they see their rule serving. Ask what they're hoping to achieve. Describe the difficulty it's causing you. Ask them for help playing your character well at their table.
(This all assumes the very best of your GM: that there's a good reason, poorly articulated, for this and that they're interested in helping you play your character. I hope that's your situation, rather than the other one: you're sitting at a table with a petty tyrant who hates spellcasters and is actively trying to make play difficult for their players. In that case I suggest you find a different table.)

A:

As in some previous questions in this vein (“Can the GM..?”), I am going to answer no.
The rules of the game put the GM in charge. Nothing specifies that there are any specific things he cannot do. But ultimately, the GM is not in charge of you, the player—he cannot force you to play. I for one would not play with a GM who insisted on this—and suggest you do the same. I wouldn’t even play if I was playing a barbarian rather than a wizard, because this rule has some major, serious problems, and even if it doesn’t directly affect me it says things about the GM that imply it’s only a matter of time before, say, I’m not allowed to apply my barbarian’s full Strength bonus unless I can personally lift what the encumbrance tables say my barbarian can.
This is not how the game works. This is not what is expected of a wizard player. This isn’t even an accurate understanding of what leveling up represents.
Now, your GM, you’ve said, is trying to keep true to the spirit of the rules. That suggests that this is a mistake, not power-tripping GM megalomania, since this is very much not in the spirit of the game at all, and your GM just seems to be a bit confused.
First, leveling up represents the accomulation of the previous level’s incremental improvements. It is the “aha!” moment when it all comes together and your new abilities become reliably usable. Thus, when you say that

His reasoning is because the character gains a new ability, the wizard gets two new spells every level, doesn't mean they can use them 100% the first couple times, which I understand.

My only response is no, it means exactly that they can now use the new abilities 100% of the first couple times.
Second, ultimately, you are playing a character who is not you—the wizard with his 18 (or higher) Intelligence, and his literally spending his entire life memorizing spells, can do things you cannot (e.g. memorize all of these spells). That is, in fact, much of the point of playing an RPG. Likewise my horribly-out-of-shape butt can play a barbarian capable of sprinting at impossible speeds for impossible amounts of time, lifting impossible amounts of weight. That’s literally the game.
Third, and perhaps most importantly, there is the matter of fun—memorizing the precise text of every ability you have isn’t generally considered to be that. Should you know generally what your abilities do, how they work and when you would want to use them? Yes, because that keeps the game moving—keeps everyone having fun. But double-checking something when it isn’t even your turn is generally considered totally fine. As is checking the exact rules when a bizarre case comes up that no one has encountered before and you need to figure it out.
This rule doesn’t allow any of that, unless you spend an enormous amount of time and effort memorizing. That doesn’t sound like how I want to spend my (generally limited) free time. It actually sounds like it’s punishing me somehow for picking a wizard.

A:

Apply clue-book to DM. Repeat until satisfactory results are obtained.
Your DM doesn't have a clue.  First — the leveling process reflects fully consolidated learning, not the first fits of learning something new.  Once your character has learned something as part of the leveling process, they've learned it, full stop — they don't need extra practice after they've leveled lest they forget it.
Second — the level of system mastery he is trying to demand is literally impossible in larger, more sprawling systems where splatbooks dominate.  People don't memorize entire spells in systems like D&D 3.5e and Pathfinder because down that road lies ruin, especially when rulings start hinging on the precise wording of spells.  It's not a good habit for a DM to drive players towards.
Third — the action economy in 5e clearly wasn't designed for player-at-the-table i.e. out of character actions to be considered within it — it's about what your character can do, not what you as a player can do.  Does this DM make you forfeit turns altogether because of bathroom breaks?
Fourth — this is a massive penalty to casting classes (vs. mundanes).  Fighters (even in 5e) are straightforward — they only have a small handful of class features to work with.  Prepared casters can change out their spellbook on a day-by-day basis, and have a massive pool of spells to draw from.  (Does your DM make the mundanes memorize their class features?)
In case another DM does this out of sheer impatience with players looking stuff up
You said you've got your spells on your phone for reference, but pawing through large amounts of text — or webpages — isn't a very fast task on phones. The GM may be impatient with that, or the players who don't remember what their characters can do, or both.
Ask your DM if you can print out and reference a paper crib sheet of your spells — it's faster to go through and easier to deal with at a table. Staple it to your character sheet if you'd like.
At a minimum, a crib sheet should have the spell name, level, magical school, casting time, range, duration, basic spell components (i.e. Verbal/Somatic/Material, and the GP value of material components that have such), and a brief description of a spell's main thrust — think "classified ad" if you have ever seen classifieds in a physical newspaper before.
Example format:

Sending  Evo3  1 action  Unlimited  V/S/M  1 round
  Send 25wds/less to known target w/ Int 1+, 5% fail for inter-planar
Shield  Abj1  1 reaction  Self  V/S  1 round
  +5 AC incl. against trigger + immunity to Magic Missile
Raise Dead  Nec5  1 hour  Touch  V/S/M500gp(C)  Instant
  Return dead creature to life w/in 10 days of death, creature returns w/
  1 HP and -4 to attacks/saves/ability checks, removed 1/longrest

