This paper addresses the issue from a US Constitutional perspective. A variant question is whether the laws or constitutions of any state actually allows such an action. The US constitution does not say that governors have any veto power whatsoever, so it is a matter of state law whether there is the potential for such a veto. The essay argues that a governor has such a power, in the situation that you describe. One case where there is no such possibility at least withing the scope of state law is Washington state, where the law requires absolute compliance with the results of the vote. That is, a faithless elector will be replaced until a correct vote is achieved, and there is no provision for "we can't decide".
W.r.t. the US Constitution, popular vote is a matter of state discretion. Every state has legal provisions stating how electors are to be selected, and how the votes of electors are to be handled. The Washington legislature could, theoretically decide that they do not like the probable outcome, and change the law so that the legislature itself votes on electors. But such a change in the law is subject to gubernatorial veto under Washington state law – and legislative override of said veto. In Washington, there is no existing law saying "If X happens, then the legislature directly decides".
Arizona State Legislature v. Arizona Independent Redistricting Commission(2015) is somewhat relevant, in that there was a matter that hinged on the word "Legislature" in the Arizona Constitution. The court interpreted the reference to "Legislature" to be to the system of lawmaking, and not specifically the senate+representatives houses:
our precedent teaches that redistricting is a legislative
function, to be performed in accordance with the State’s
prescriptions for lawmaking, which may include the referendum
and the Governor’s veto.
It should be noted, though, that Roberts etc. dissented, and the demography of the court has changed since then. At any rate, the article concludes that it is unlikely that SCOTUS would overturn a gubernatorial veto, on the grounds that the government is not a mamber of the "legislature" in one reading of the term.