If I’m ace and sex repulsed and would never have sex with my romantic partner, am I obligated to let them date and/or sleep with other people?
You’re never obligated to do anything for a partner…but then, no one is obligated to do anything for you.
If your terms are “dating me means being monogamous, which also means no sex ever,” there are lots of people who would not be okay with those terms. So don’t date those people!
If anyone makes you feel obligated to do something in a relationship that you’re not comfortable with, that’s not a healthy relationship. But the flip side is true too: if you feel that your partner is obligated to forgo sex entirely to date you, and your partner is not comfortable with that, it’s not going to work out.
It seems your best bet is to find another person who is OK with a monogamous romantic relationship that doesn’t involve sex. Or, you could try doing the self-work necessary to “let” your partner see other people without it feeling like an “obligation,” but again, doing that self-work isn’t an obligation either.