Does the FAA prescribe radar cross-section requirements for civilian aircraft? Are there any FAA regulations stipulating the minimum permissible radar cross section for civilian aircraft (i.e., making sure that civilian aircraft are at least such-and-such cross-section visible to primary radar)?
No. Primary radar isn't a universal tool to begin with. Secondary radar and thus transponders are more commonly used. So there are regulations as to where and which aircraft need to install transponders. Ultralights and some GA can get away without.

Related: Is it legal to make a non-military stealth plane?

No RCS requirement, but you'll need a transponder regardless.