Does Federal law require states to bargain with public sector unions?

So, a conservative dream (which I also hold) would be to bust the public sector union cartels. The most direct and achievable way to do this WOULD be with state-level referendums, however I THOUGHT that federal law REQUIRES the states to deal with the unions. That is, even if 90% of the populace of a state voted to ban collective bargaining as it is now with the public sector unions (probably to replace it with being controlled by state laws which the people could vote for), the Supreme Court would shoot the law down as violating federal law.

However, doing some research, it seems like this isn’t so.
There are federal union laws (duh), but at first they didn’t apply to public sector unions. From there, it was all executive orders. One in New York state, and one Federal by JFK.

It doesn’t seem like a referendum that busted the public unions would run into much federal interference.

Does anyone know where it may run afould of Federal law, ignoring the larger questions of federal power/federalism which would be yet another challenge later on that would have to reach SCOTUS?

“OMG Edwin, I’m some guy and I totally know about federal laws and I can tell you all about that”

Gee, thanks guy