Do US universities validate your degree before making an offer? I have received a verbal offer for a faculty position in the US. This was 3 weeks ago when the chair of the search committee called me to tell me the good news in addition to offered salary, etc.

As of today I'm still waiting to receive anything in writing and, upon asking, I was told that they cannot send me the written offer until they have validated my degree, which they said might take an unknown amount of time. (Note that I've obtained my doctorate from a French University that has been around for 500 years and is accredited on all continents.)

I noticed on the department's website that there are still seminars being scheduled that may or may not be talks by faculty candidates. Meanwhile I have turned down another offer and I'm beginning to get a little stressed about the uncertainty. 

Thus my questions: 

1. Is the above the normal process for hiring people with non-US degrees?

2. Is there anything I can do to reduce the uncertainty?
Some universities do, some don't. For those that do, it is not because of suspicion on the part of the search committee that you don't really hold a degree. The academic world is small enough that we rely on mutual trust for many things: presumably the committee members know some of the people who wrote your letters of recommendation. However, especially for public institutions, there are often various rules and regulations that must be followed, paperwork to be filled, and other details to process. The fact that you have a degree from a foreign university is probably slowing things down, as the admin staff may not be used to dealing with it. On se débrouille.
If you really like, you can email the search chair, asking if there have been any updates, as you are anxious to get a firm offer in hand. Also, remember, a verbal offer is worth the paper it is written on. Until you have signed a contract, you are still on the job market and you should be acting as such (e.g. applying for jobs, attending interviews, etc). However, as you've said, there's a lot of uncertainty. So... Relax. Take a walk. Write that paper you've been procrastinating on. Enjoy the long French evenings.