Both the primary election and the general election are government-organized events. These elections are primarily established and regulated by state legislatures.
Political parties can choose not to participate in a state primary election. But if they chose to take part in one, they are subject to all Constitutionally valid state laws regulating the election.
Parties have certain First Amendment association rights that election laws must not infringe, but I don't think that is relevant in this context.
I wrote a longer post with more details as an answer to a broader past question here: How do parties really choose candidates in the US?