Using your $X$ and $Y$, you get an augmented simplicial object as follows: $$ \cdots Y \times_X Y \times_X Y \Rrightarrow Y \times_X Y \Rightarrow Y $$ Applying $\cal F$ to this diagram, you get a coaugmented cosimplicial ring spectrum. The spectral sequence for the homotopy groups of Tot of this which realizes your Cech cohomology.
So this reduces you to a question: Given a cosimplicial object in ring spectra, do you get a multiplication on the associated spectral sequence converging to the multiplication on Tot?
This is true; most types of multiplicative structure carry over like this (although it may turn into "coherent" multiplicative structure). However, I've had a little trouble chasing this through the literature this morning.
One method you could use is a method of universal example, where $E_r$-cycles are carried by certain maps of cosimplicial objects. In Bousfield and Kan's "A second quadrant homotopy spectral sequence," they do this for the smash product in the homotopy spectral sequence of cosimplicial spaces (which is harder because you have to worry about basepoints!), and other authors (James Turner, and recently Philip Hackney) have studied operations arising on cosimplicial chain complexes by methods that should translate to the context of ring spectra.
Wish I had a more definitive reference for you.