Meghan, an American, says in the video that every four years elections are referred to as the most important election of our lifetime. “But this one is,” she stressed. “When we vote, our values are put into action and our voices are heard.”
Harry called on Americans to “reject hate speech, misinformation and online negativity”.
Neither mentioned Mr Trump or his Democratic challenger, Joe Biden, by name but some have interpreted their comments as critical of the president.
Mr Trump was asked about the video during a White House press conference, with the reporter saying the couple “essentially encouraged people to vote for Joe Biden”.
“I’m not a fan of hers and I would say this, and she probably has heard that, but I wish a lot of luck to Harry. He’s going to need it,” Mr Trump replied.
The couple recently moved into a California home after announcing they were quitting royal duties and moving to North America, citing what they said was unbearable intrusion and racist attitudes from the British media.