Can I be forced to provide DNA test?

I'll try to make this as short as possible. I began "casually" dating a man that swore to me he had nothing to do with his ex other than they shared a son. Soon after I became pregnant from him. We lived together during my pregnancy, (during this whole time he helped me financially, i met his family, etc) and on my 8th month of pregnancy he left me and and it seemed he "went back with his ex" (although at the time i couldn't prove it.) When my child was born he legally acknowledged him as his son as well. He never mentioned any doubts about paternity. A couple of months ago, his "ex" who happened to be his girlfriend all this time (she would live out of the country for 6 months & 6 months in the US) just found out about his infidelity and his new son. He never even mentioned he had another son to her. Both me and his girlfriend confronted him, he denied being the father of my son to her even though she saw the birth certificate stating he was my child's father. I then asked him in front of his girlfriend if he wanted DNA test and he said no. Two days later he said he did want DNA test because he suspects my son is my ex boyfriends (who I had recently broken up with before dating my son's father). Deep inside I know he asked for DNA only to convince his girlfriend I was just a fling and a liar. (He even denied ever living with me or introducing me to his family) He humiliated me by denying and rejecting my son and requesting a pointless DNA test. I told him I did not want his money or any type of support for my son after what he had done. Now he keeps requesting a DNA test. It's been almost 2 months where he hasn't even given me $ for diapers or formula. I had to move back with my mother for support since he left me with no financial support after I told his girlfriend the whole truth he had kept from us for 2 years. (Knowing I just lost my job and have no income to support my son) I want to know if he can legally claim rights to my son (shared custody, etc) against my will since he is the legal & biological father of my son. Can he force me to provide DNA test? I don't think a man that rejects his own son & doesn't even provide for diapers should have the right to spend even 1 minute with my son. Is there a way I can change my son's last name to mine only? Thanks to those who took the time to read all this and answer my question. Blessings.