Group signatures (GS, Chaum and van Heyst, EUROCRYPT 1991) are digital signatures that allow a signer to anonymously prove the membership and also allow the special authority called the opener can identify the signer. Group signatures with message-dependent opening (GS-MDO, Sakai et al., Pairing 2012) weakened the power of the opener by introducing another authority called the admitter who issues a message-dependent token. It would be a natural research topic to clarify whether cryptographic primitives that are required to construct GS-MDO are stronger than those of GS or not, according to the enhanced functionality of GS-MDO. In this paper, we propose a generic construction of timed-release encryption (TRE) from GS-MDO. Note that Sakai et al. have shown that GS-MDO implies identity-based encryption (IBE), and Nakai et al. (IWSEC 2009) and Matsuda et al. (Pairing 2010) demonstrated generic constructions of TRE from IBE. Thus, we do not show any new result from the viewpoint of feasibility. We show that (1) GS-MDO directly implies TRE without employing the generic constructions of TRE from IBE, and (2) the proposed TRE construction provides public verifiability, that is not usually supported by TRE, because a TRE ciphertext is a group signature in our construction. Our result implies that GS-MDO is a very strong cryptographic primitive.