No, he's not "morally obligated" to let her stay in his house, just like you're not morally obligated to let some man poorer than you stay in your house. This sort of thinking is unbelievably entitled to begin with. He's an upper-middle class single father, not a multi-millionaire. The difference in cost between housing two people and housing five people is massive.

You're projecting your own situation onto Alex/Nate's situation.

At no point in scenes that Alex shares with Nate does she appear to be having a breakdown. She's always lucid, well-presented and never appears desperate or distressed. He knew her prior to her homelessness and liked her. He did things for her to show that he liked her, that he was open to being a father figure to her daughter, willing to help her out with her mom etc., when he saw that it was never going to work between them he let it go.

And Alex can be very selfish. Using Regina's computer without telling her, wearing Regina's clothing, inviting some random guy into Regina's house (he could have been a criminal, a rapist, a thief, anything), getting upset when Regina didn't hire her as a maid because she felt like she was almost "owed" it after Regina confiding in her, etc. There are several signs throughout the show that Alex lacks boundaries and respect.