I'm Going Through A Divorce And I Have A Daughter That Is Almost 2. I Am Seeing Someone Else Now But My Ex Doesn't Approve Of Him Because Of His Past. He Says I'm An Unfit Mother And Is Going To Take My Child From Me. There Is No Markings Or Bruises On My Child. I've Never Put Her In Harms Way And She Has Never Been Hurt In My Care. What Can He Do That Will Prove And Will Make The Courts Believe I'm Unfit?

It isn't the parent that has a right to their child but the child that has rights to their parents. As long as you have the child's best interests at heart there isn't anything he can do legally. Some things that are considered: Health, finances, environment, etc

Anything that he alleges by law they can investigate you and if they say what he has said is unfounded then there is nothing he can do. He CAN file for custody and then it is up to the Judge to decide which parent the children are better off living with.

I'm Going Through A Divorce And I Have A Daughter That Is Almost 2... What Can He Do That Will Prove And Will Make The Courts Believe I'm Unfit?
You have a lawyer, right? Your lawyer can find out everything he plans to present as evidence of your unfitness.

Unless he has proof of child abuse or neglect, he really cannot do anything. The worst he could do is contact CPS, they may do an investigation (including speaking to friends/family) They will look at your living conditions, ask about your work, interview the people that live in your home, etc. But again if they cannot find any proof of abuse or neglect, they cannot do anything.
Just watch your back, no telling what he may or may not do to keep the children away from the new boyfriend (lies, etc.)