Because of McConnell's own fuckery with Kentucky law regarding filling vacant offices, in 2021 and again in 2024, the legal question is unclear. Gov. Beshear has been very open about it lately, saying, "We're looking at everything," when pressed about what legal option(s) he may take.
To answer your question directly, he could try to get a judge to rule requiring McConnell's presence or personal answer, but that would require being able to demonstrate a legal necessity on behalf of the state, which seems self-evident but which he may not actually have in law. During the first month he was also been reluctant to unduly publicize McConnell's "private health" issues, but his patience is gone and he is no longer holding back.
It's also pretty clear that the governor could simply declare the seat vacant, pointing to McConnell's prolonged absence as proof and call the special election anyway, just to force the matter, but that's not as simple as it seems either, because even if a vacancy becomes apparent, with or without McConnell's cooperation or disclosure, the legal question is far from settled.
From before Kentucky was a state (it was part of the commonwealth of Virginia under royal charter), the governor had the right and the power to fill vacant offices, and did so without issue. Upon statehood, that right of the governor to fill vacant seats with the candidate of his choosing was codified into the Kentucky state constitution. Until recently, a federal seat has been included in that right and power.
But in 2021, when McConnell's health started to fail, he had the Kentucky state legislature enact a law that required the governor to select his own replacement from a shortlist of three candidates supplied to him by the state legislature.
Apparently that wasn't good enough for McConnell, and in 2024 he had the Kentucky state legislature change the law again, this time requiring that the governor make no replacement choice for his own seat at all, but rather call a special election -- unless that vacancy occurs within three months of a regular election, in which case the seat will just remain vacant until the candidate elected in November fills the vacancy in January. That's what the August 3 date people are throwing around is about: it's the three-month-prior to the regular Kentucky election date in November.
The problem here is that none of these recent laws have been tested, AND they directly contradict the Kentucky state constitution: meaning that whatever the governor does, there will be legal challenges.
Meaning that all these people in this thread calling out what they think is the obvious stupidity/ignorance of other people in the thread are exactly that themselves, because they have missed the most obvious thing of all and what the governor himself has been pointing out repeatedly: literally NO ONE knows what will happen now.
And that is likely what McConnell meant to accomplish: all along he wanted to ensure that should his health fail, a Dem governor could not appoint his interim replacement, and legally that is all but certain just because no matter what happens, the legal challenges will likely run out the clock.