There is no prohibition against doing that on a tourist visa. No, you cannot file for him to stay in the U.S. if you are not married. He may be eligible for a longer tourist visa based on your domestic partnership.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
If you establish domestic partnership, he will be able to come to the United States with you on a B-2 tourist visa. He will be able to request 6 month extensions of the visa as long as you are in the United States, but a B-2 does not provide employment authorization. A B-2 also requires nonimmigrant intent, meaning he will have to demonstrate that he is not planning on coming to the United States permanently.
This email is for general information purposes only and does not create an attorney-client relationship.