Kentucky Gov. Andy Beshear called on Sen. Mitch McConnell to publicly demonstrate that he remains capable of serving or resign, a day after McConnell's office said the 84-year-old Republican senator was still not medically cleared to leave rehabilitation.

"As Governor of the state you serve, I am calling on you to directly and verbally address the people of Kentucky and provide proof of your capacity to serve, or resign," Beshear, a Democrat, wrote Monday.

Beshear said McConnell could do so by releasing a short video or calling a Kentucky radio or television station. McConnell on Monday released a still photo of himself with his wife, the second time doing so since he was hospitalized.

"All we're asking is just a little bit of proof," Beshear told reporters Monday. "When you run for these jobs, you know you're giving some of that privacy up."

Beshear also asked Senate Majority Leader John Thune, R-S.D., to investigate McConnell's condition if the senator does not show that he can continue serving.

McConnell's office did not immediately respond to CNBC's request for comment.

His absence leaves Republicans with one fewer vote in their narrow Senate majority and raises questions about Kentucky's untested process for filling a Senate vacancy.

Beshear cannot remove McConnell, who is not seeking reelection when his term ends in January. A vacancy would occur only if McConnell resigns, dies in office or is expelled by a two-thirds vote from the Republican-controlled Senate.

Kentucky's Republican-controlled legislature passed a law in 2024 barring the Democratic governor from appointing a temporary senator and requiring a special election. The law has never been tested, and the seat would remain vacant until a winner was certified.

Legal experts have said the law could face a challenge because the Kentucky Constitution says vacancies in statewide offices "shall be filled by appointment of the Governor." That language could give Beshear grounds to argue he retains authority to name a temporary replacement.

The timing is increasingly important because Sept. 1 is the practical deadline for holding a special election on Nov. 3, according to Kentucky law.

McConnell was hospitalized after a June 14 fall that briefly left him unconscious and later developed what his office called a mild case of pneumonia. His office said Monday that he was undergoing intensive physical therapy and released a new photo of him with his wife, former Transportation Secretary Elaine Chao.

Beshear said those updates did not establish whether McConnell could perform his duties.

"Even the attending physician's written statement does not discuss your actual condition, including whether you can speak, reason, or carry out your duties as a senator," he wrote.