The governor might run into issues if he tried this. The Constitution explicitly delegates the right to regulate commerce with Indian tribes to the federal government. If a state government tried to interpose itself between the federal government and a reservation in the manner you described, it would be easy to cite federal supremacy.
Oklahoma is furious that the conservative controlled Supreme Court (in an opinion lead by Gorsuch no less) ruled that the government in Oklahoma in fact didn’t get to trump treaties the Native American tribes had signed with the Federal government.
There is, of course, virtually no path to disestablishing the reservations via Federal legislation.
Like I said, it likely wouldn’t be legal.
But if you put aside the issue of the law, then it comes down to an issue of who can bring more force to the dispute. And a state isn’t going to be able to stand up against the federal government. Orville Faubus learned that lesson.
The governor of a state can’t just arbitrarily block utility and transport connections anywhere in their state, let alone around Indian reservations. There are tons of actual statutory laws, contracts that have been signed, property rights implicated et cetera. Shutting down roads and such civil authorities including the state Governors often can do that within certain contexts, but if they try to do it just for no reason they can get in a lot of trouble. The Governor of a State doesn’t rule his State as a king, and can’t get away with just deciding to order a bunch of crazy things. He would immediately lose in court. If he chose to ignore the court order, his orders to his underlings would not be legal–if they continued to obey the Governor instead of the court, the State employees enforcing the Governor’s order would be subject to arrest and prosecution for contempt of court among a number of other things. The Governor is very unlikely to have that many loyalists in State government willing to potentially lose their civil service jobs and their freedom for his political stunt.