  In this article, we prove an explicit Waldspurger formula for the toric
Hilbert modular forms. As an application, we construct a class of
anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms,
generalizing the construction of Bertolini, Darmon and Prasanna in the elliptic
case. Moreover, building on works of Hida, we give a necessary and sufficient
condition when the Iwasawa mu-invariant of this p-adic L-function vanishes and
prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg
L-values with anticyclotomic twists.
