We have an employee who resigned with 7 days notice. We accepted his resignation as of that day. Do we need to pay the 7 day notice he gave?

It would be proper to pay him, He acted in good faith, giving notice, You would not allow him to work this notice, so you need to pay him for this notice.

I ran into this situation many times and we would typically pay a person through their notice if it was our decision to terminate employment upon receipt of resignation notice. Technically, if your state is an "at will" state you do not have to however as Fr_Chuck pointed out it's the right things to do.