Senator Mitch McConnell, Republican of Kentucky, broke his silence on Sunday about his long hospitalization, releasing a statement saying that he was recovering at a rehabilitation center after suffering a fall and losing consciousness last month, and then battling a “mild” case of pneumonia.

“My doctors have confirmed that I didn’t break any bones or suffer a concussion,” Mr. McConnell said in the statement, which included a photograph of him holding a newspaper published on Sunday alongside his wife, Elaine Chao. “I didn’t have a heart attack or a stroke. I don’t have any tumors or hemorrhages.”

Mr. McConnell said he had been taken by emergency responders to the hospital after falling “briefly unconscious.”

The 84-year-old senator’s update regarding his health came on the heels of Senator Lindsey Graham’s sudden death on Saturday, which prompted a fresh round of scrutiny about the Senate’s aging population. For weeks, Mr. McConnell, who is more than a decade older than Mr. Graham was at the time of his death, did not share details regarding his health ailments despite multiple requests from The New York Times and other media outlets.

Questions had mounted over what prompted his lengthy hospitalization, what his condition was and whether he would ever return to the Senate. Emergency responders were heard in dispatch audio reporting that they were performing CPR on an unconscious individual suffering cardiac arrest at the senator’s Washington address last month on the morning he was hospitalized. Scant statements from his office only fueled rampant speculation across Capitol Hill and the country regarding his medical status and whether he would ever make it back to the Senate floor.

Some began openly wondering what would happen should Mr. McConnell, who is retiring in January, be unable to finish his term in the Senate. Gov. Andy Beshear of Kentucky sent a letter to him demanding that he update his constituents on his health as rumors swirled, questioning whether he was still fit to serve.

Mr. McConnell said in his statement that he had been reluctant to share his physical challenges, which have plagued him since he contracted polio as a child and have grown more prevalent with age, contributing to a steep decline in recent years.

“You all know how folks of my generation often hesitate to share the vulnerability that comes with growing older,” the former Republican leader said. “Even in the public eye, I feel that same instinct — I can’t help it.”

Last week, prominent Republicans said that they had spoken with Mr. McConnell, in formulaic and seemingly scripted social media posts and statements that described “lengthy” conversations with the Kentucky senator, including about key issues that are top of mind for conservatives. Rather than putting it to rest, the statements only further intensified already wild public speculation about Mr. McConnell’s health.

In his statement on Sunday, Mr. McConnell said he had left the hospital and that he was entering a physical rehabilitation center to help him regain strength after the fall. He did not provide an estimate of when he would return to work.

Attached to Mr. McConnell’s statement was a comment from the Office of the Attending Physician, the on-site doctor for members of Congress at the Capitol.

The Capitol physician’s office said the senator had fallen because of his post-polio condition, sustained “minor injuries” and was now undergoing “intensive physical therapy.”

Mr. McConnell turned to the same physician, Dr. Brian P. Monahan, in 2023 when he was working to dispel speculation that he was no longer physically able to continue as minority leader after a series of on-camera freezing episodes that medical experts said could be caused by a stroke or seizure. Back then, he released a letter from Dr. Monahan declaring that an examination and tests had ruled out a stroke or seizure as causes.

“There is no evidence that you have a seizure disorder or that you experienced a stroke, T.I.A. or movement disorder such as Parkinson’s disease,” the letter said, using a shorthand for transient ischemic attack, a kind of mini stroke.

The statement on Sunday similarly ruled out the widely speculated potential causes of Mr. McConnell’s recent episode.

“A comprehensive evaluation by a multidisciplinary team determined that he had no fractures, cardiac abnormalities, stroke, tumor or hemorrhage,” the physician’s office said.