“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”.
In response the US President said: “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.”
In the clip for the 2020 TIME 100 list Harry pointed out that he cannot vote in the November 3 showdown.
Senior royals are not allowed to vote in British elections. Traditionally, members of the royal family are expected to be politically neutral but Meghan has confirmed she will vote in the US poll.
Piers Morgan also criticised the royals for sharing the political statements as he said it showed their “brazen disregard for royal neutrality”.
The presenter wrote in a column for the Daily Mail: “By expressing themselves in such a brazenly partisan manner, they’ve crossed a massive line which should now have serious consequences for the couple who specialise in wanting their royal cake and the freedom to eat it.”
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.
Additional reporting by Associated Press.