Former New York City Mayor Rudy Giuliani (R) is a big fan of Donald Trump – and he's said some strange things while campaigning for the Republican nominee. (Peter Stevenson/The Washington Post)

WE ARE a little worried about Rudy Giuliani, the Republican former mayor of New York. Is “America’s mayor” okay?

During his 15-minute speech at the GOP convention last month in Cleveland, it was notable that when he said Donald Trump loves “all people, from the top to the bottom,” Mr. Giuliani animatedly gestured toward his knees as he said “top,” and above his head as he said “bottom.” Also, why did he say that he and his wife, Judith, have been friends with Mr. Trump for 30 years, though he met his wife in 1999, only 17 years ago?

Also — we’re noting this purely out of concern — during his speech he often licked his lips, indicating dry mouth, which, according to the Mayo Clinic, can be a symptom of nerve damage, stroke or Alzheimer’s disease. At the end of his address, beads of sweat were visible on his pate — did that not suggest heart disease?

Mr. Giuliani is just 72, but he seemed slightly stooped as he walked to the lectern, where his wide stance made us wonder if he’s unsteady on his feet. Then there was his slurred diction, as when he referred to “jushtified” police shootings and Syrian “refyoongees.” More evidence of a stroke?

Yes, all of the above is ludicrous — as ludicrous as Mr. Giuliani’s own loathsome smears and innuendo about Hillary Clinton’s health, which follow Mr. Trump’s. In recent days, indulging a grudge he has evidently held for years, he has urged people to watch Internet videos that purport to prove Ms. Clinton is ill. He doubled down after that, saying the Democratic presidential nominee appeared “tired” and “sick.”

Former New York mayor Rudy Giuliani glossed over the 9/11 attacks when speaking at a rally for Republican presidential nominee Donald Trump in Ohio on Aug. 15. "Before Obama came along, we didn't have any successful radical Islamic terrorist attack in the United States." (Reuters)

Of course, Mr. Giuliani hasn’t a speck of evidence for his blather, a damning fact considering he was once a federal prosecutor, and proof positive of his impaired character, if not health.

Voters have a right to relevant facts about both presidential candidates’ health and are correct to question the degree to which each (or, more accurately, neither) has been sufficiently forthcoming about his and her medical history. In fact, Ms. Clinton has released more (and more relevant) information than Mr. Trump, whose own physician, Harold Bornstein, a gastroenterologist, quackishly asserted that, “If elected, Mr. Trump, I can state unequivocally, will be the healthiest individual ever elected to the presidency.” (One can only imagine the detailed study Dr. Bornstein has devoted to every previous president’s health, including that of Theodore Roosevelt, an accomplished boxer, hunter and outdoorsman who, when he became president, was 28 years younger than Mr. Trump.)

Unfortunately, Mr. Trump, 70, has taken the same nondisclosure stance on his health records as he has on his taxes, asserting there is nothing to hide while hiding everything. Ms. Clinton, by contrast, has released some test results, which, as far as they go, indicate good health.

That hasn’t stopped Mr. Giuliani from trading in scurrilous and debunked theories about the Democratic candidate. Come to think of it, he should see a doctor.