No. Producing a usable vocal sound, and reading or playing music are separate skills. There are plenty of good (and well employed) pop singers who don't read music, or read at an elementary level. Reading ability is useful, however, for every singer, and necessary, to some degree, for classical and musical theatre. But, if your time is limited, and you really want to sing, better to start with that and not get hung up on things you think you "ought" to do. Playing an instrument, especially piano or guitar, is very useful for developing one's ear, choosing keys for songs, learning melodies and harmonies, etc., but is not necessary in learning to sing.