Not in the United States. "Marriage" means that two people intend to live together permanently. "Starting a family" means deciding to have children.
(Yes, a married couple without children can also be called a "family." A person might say "I quit my job because I wanted to have more time with my family" when he has a spouse but no children. It's confusing.)
Marriage means that the couple is recognized legally as a couple, and gives them special legal rights. For some people it is also a religious ceremony, and one Protestant tradition says that it is "intended by God for their mutual joy; for the help and comfort given one another in prosperity and adversity; and, when it is God's will, for the procreation of children."
Many people marry with no intention of having children. Many people marry long before they decide to have children. Some people marry only when it becomes clear that they are about to have children.
In the United States, some people marry "in judge's chambers" without a religious service. It is common for couples to have a "church wedding" in which the religious and legal aspects are combined. The clergyman is authorized by his or her church to perform the religious ceremony, and also holds an official legal position that gives them the authority to perform the legal marriage at the same time.