I just have a month to month room mate who only within 2 weeks of stay started violating the terms of our agreement. I therefore decided to terminate the contract. Can I evict the person without involving the court. In other words what is the minimum days requirement (in NYC)for this kind of tenancy to bcome legally binding?

It sounds like this agreement was written. In that case it doesn't matter, 1 day is enough.

You can give him a notice saying that it isn't working out and you want him out by the end of next month. Only if he doesn't leave do you need to go to court to evict.

It is a verbal agreement not written and we had a witness present. However the room mate made one month payment and a deposit.

That should be enough for him to prove, in court if necessary, that you had an agreement. So you need to give him one rental period's notice.