Ray Cole, from Deal, Kent, was detained with his Moroccan partner by police who approached the pair at a bus stop in Marrakech last month, his family said.
His son Adrian Cole said the retired magazine publisher, 69, was jailed for four months on Thursday.
Dover and Deal MP Charlie Elphicke warned travellers that Morocco was unsafe for British tourists.
The Conservative MP told the Pink News: “I have been doing all I can to help free Mr Cole from these appalling charges.
“I urge people not to visit Morocco. If you go there you are at serious risk of facing trumped up charges for medieval crimes.
“The message is clear: Morocco is not safe for British tourists.”
A #freeraycole hashtag has been set up on Twitter and a Facebook page has also been set up calling for Mr Cole’s release.
His family claim his health is at risk due to the conditions in the city’s central prison.
Foreign Office guidelines state that homosexuality is a criminal offence in Morocco and sexual relations outside marriage are also punishable by law.
A Foreign and Commonwealth Office spokesperson said: “We can confirm the detention of a British national in Morocco.
“We are providing consular assistance.”
The Moroccan Embassy has been contacted for comment.