Discussing work with supervisor in an invited dinner with his family I am a grad student. I have been away for an internship in another city. My professor came to meet me at the weekend and he invited me to dinner.
I did not have any time to prepare and I did not bring a gift and he brought me a gift. I was embarrassed, really.

I recently got a result from my project and was very interested to show him my result as I had not seen him for a long time.  I am feeling so embarrassed.

Do you think these behaviors were rude? I was very clumsy. Could I write something as thankyou letter and apologize about not being prepared?
Welcome to academia.SE :-)

See, many things that we do in our life are most of the time unplanned and spontaneous. The thing that you did is not a big deal for good people. Just say sorry sometime during tea time and get on with it. Don't keep it scratching.

Further, one positive to that is: He would have felt so happy that he has got a very dedicated (and weird) student who shows results during family dinner.

I have done it quite few times and I am not sorry about it. It feels weird initially, but not so apologetic. 

Good luck!