Yes, I do believe the Internet cuts into family time. Anything that is not family time cuts into family time by definition, so I do not really understand why this question is even being asked. If something is not family time, then it is clearly cutting in to family time.
The internet has changed the way that people socialize by allowing people all over the world to communicate instantly, but this hasn't really changed the fact that people socialize outside their families, it just changed the way that socialization is performed. Families continue to communicate in person if they live together, and with the internet now extended families that don't live near one another are able to stay in better communication through email and social networking sites.