Louis Arthur Charles was born on April 23, 2018, at St. Mary's Hospital in London.
The Queen was born Elizabeth Alexandra Mary in London on April 21, 1926.
The Prince of Wales stepped out for his first in-person engagement after the Princess of Wales shared her health update on March 22.
"Said he wrote a book," the actor quipped after running into the Duke of Sussex.
The Netflix show ended its six-season run last year.
The royal appearance comes a little over a week after the Princess of Wales revealed she'd been diagnosed with cancer.
Buckingham Palace previously announced on February 5 that King Charles had been diagnosed with cancer.
The Princess of Wales shared the health news in a video statement on Friday.
Charles Spencer shared a message for the Princess of Wales after she released a video discussing the status of her health.
The Princess of Wales said cancer was detected after her surgery earlier this year.