No, the Church does NOT need the authority of the state in order for two people to marry validly. However, it does appear to be more prudent to make sure that the couple is married in both the eyes of the Church and the state.
For example, according to the Code of Canon Law, a woman can marry validly once she turns 14 and a man can marry validly when he turns 16 (without parental consent!). In most Western countries, this generally doesn’t happen very often, but the Code governs the universal Church, not just one country or a few countries. The Church does suggest for national bishops’ conferences to encourage people to marry at the recommended age suggested by the state as well.