Congrats, dude! Nothing more special than newborn time. Having just gone through the newborn stage months ago, I can vouch just how unique of a feeling it is. You just don't realize how happy you can be while at the same time stressed to the brink of insanity.
Speaking on all video gaming in general, here is my experience:
I seemed to find that I played certain video games when my kid was a newborn, but it was completely gaming wrapped around the situation. When I had to stay up at 2 in the morning trying to get my kid to sleep, I sometimes played bejewelled and rocked him in the chair. It was a game with no time limits, no attention needed. It was just something to do to pass the time.
When I found the time where I needed to wind down for ten minutes, I would often play an arcade game. I didn't have the time and energy to invest anything real for video gaming, so it was just a quick game here and there. If I only had a couple minutes, I would play a game of robotron.
When my son developed a regular sleep schedule, I would spend some time before going to bed myself to play games, maybe an hour.
When my son got to the age where he could crawl around, I found much more time to play video games for a length of time. My kid was more interested in moving and shaking, and he didn't mind if my attention was not always focused on him anymore. Of course the flip side is that my son is getting much more fun to play with each day, so playing with him often outranks video games.