This is sort of interesting. In Virginia, incumbents are allowed to pick the process by which nominations for their seat are made. A part of that law was struck down. There is also no party registration. The 4th Circuit said that the law is unconstitutional, as applied, when the incumbent picks a primary as the nomination process:
The U.S. Court of Appeals for the 4th Circuit handed down a ruling today in the case of Miller v. Brown affirming in full the decision by Judge Hudson of the U.S. Dist. Court for the Eastern Dist. of Virginia that the Commonwealth’s open primaries law is unconstitutional (at least with regard to instances where an incumbent selects a primary as the means of nomination and the political party committee is forced to use a method by which it is prevented from excluding voters with whom they do not want to associate.)
The appellate decision goes through a list of U.S. Supreme Court decisions on this issue and makes clear that a state must have a compelling interest for it to interfere in a party’s political nomination process and that is a high burden for a state to meet. However, in Virginia a party can use means other than a primary to select a nominee and those means can be tailored to exclude non-party members. It is only when a committee is directed by an incumbent to hold a primary and that primary must be open in accordance with state law that the committee’s First Amendment rights of association are violated.
It seems that the court is saying that one primaries require party registration. That seems like a lot to find in the Constitution.