The constitution is not a reason the two-party system developed in the United States.
A. The constitution
Although political parties play an important role in U.S. democracy nowadays, the Constitution says nothing about them. This is in part because at the moment it was written, there weren't any parties but small factions and regional blocs of people who had the same ideas on how to conduct the country. The two-party system developed at the end of the 1780s when the Founder Fathers of America utterly disagreed on ideology on what government should do and how to go about doing it (the Federalists and the Anti-Federalist, also called Republicans).
