Closed simple curves in $\mathbb{R}\mathbb{P}^2$

Well, if you take the double cover, under your assumptions the lift is two simple closed curves in $S^2,$ the complement of which will be two disks and an annulus, so the original curve bounds a disk on one side.

If C is null-homologous, then the complement of C has two components: a disk and a Möbius strip (as one sees since the preimage of C in the 2-sphere is 2 disjoint Jordan curves). If C is not null-homologous, then the complement of C is a single disk.