A very important component of character and integrity - especially in the context of leadership - is that a person of integrity acts the same way regardless of who is watching and listening. When Trump keeps saying "locker room talk", he means "things I said when I thought I wouldn't get caught." 
This is no different from someone who tells racist jokes after making sure there are only white people around to hear. It's the same as ridiculing someone's disability behind their back.
The important point here actually isn't whether Trump actually did the things he so childishly boasted about (although if he did grope and kiss people without their consent, he should be charged appropriately). The point is that our President must uphold the American principle of equality. He must make decisions based on what's best for all Americans. Clearly, his concept of respect for women means covering up his real feelings of contempt for women in public, while revealing his true attitude in private.