I think the map $F$ is injective, but not surjective. 

There is a unique map of the pants to the complement of 3 slits in $\mathbb{RP}^1\subset \mathbb{CP}^1$. If we parameterize the slits as $[a_0,a_1],[a_2,a_3],[a_4,a_5]$,
then the configuration is invariant under complex conjugation, which is a hyperbolic isometry. So the seams are the intervals $[a_1,a_2], [a_3,a_4], [a_5,a_0]$ taken in cyclic order on the circle $\mathbb{RP}^1$. The rings $R_i$ are the complements of the slits $[a_i,a_{i+1}]\cup [a_{i+3},a_{i+4}]$, indices taken $(\mod 6)$. The points are uniquely determined by the modulus of $R_i$, up to the action of $PSL(2,\mathbb{R})$, or by the [cross-ratio][1] $[a_i,a_{i+1};a_{i+3},a_{i+4}]$. Therefore the modulus of each ring determines uniquely another ring $\hat{R}_i$ which is the complement of the slits $[a_{i+1},a_{i+3}]\cup [a_{i+4},a_i]$. 
Then $R_{i+1} \subset \hat{R}_i$, $R_{i+2}\subset \hat{R}_i$. So the moduli of the rings $R_{i+1}$ and $R_{i+2}$ are bounded by the modulus of the ring $\hat{R}_i$, which is uniquely determined by the modulus of $R_i$. So this shows one cannot achieve all possible triples of moduli, so the map $F$ is not surjective. 


  [1]: http://en.wikipedia.org/wiki/Cross-ratio