Images of Andreas Hvid and an unidentified woman are posted on his website and were included in a YouTube video he shot while climbing the monument.
The video is no longer posted on his YouTube page but others have reposted it, and the footage shows the Dane and the woman reaching the 455ft top and looking out over the view of Cairo below.
An explicit still picture which is inserted into the video and posted on his website shows Mr Hvid on top of the woman who is lying on the summit of the 4,500-year-old pyramid. Neither are wearing clothes.
Another clip in the video shows the woman taking off her top.
The Egypt Today website says Mr Hvid wrote in the description for the video before it was apparently taken down: “In late November 2018, a friend and I climbed the Great Pyramid of Giza (a.k.a. Pyramid of Khufu, Pyramid of Cheops).
“Fearing to be spotted by the many guards, I did not film the several hours of sneaking around at the Giza Plateau, which lead up to the climb.”
State news agency Ahram Online said on Friday that the minister of antiquities Khaled el Anany had referred the case to the prosecutor general for investigation.
He said the action was a violation of public morality and the incident and video would be investigated by the attorney general.
Climbing the Pyramids of Giza beyond a certain level is forbidden and the whole site is out of bounds after 5pm, after which it is patrolled by police officers.
The mostly-Muslim country regards the pyramids as among its most important monuments.
The site’s archaeological director, Ashraf Mohi, had previously said he believed the video was fake because of an unexplained “illumination” near the horizon.
The pictures and video are said to have been shared hundreds of times on social media, with an angry reaction from many of those who commented.
One of the latest posters, Martin Eiler, who commented on Facebook after the image had been removed from that site, said: “Andreas; think about and remove your pyramid (picture) from your website. It’s not legal, but what’s worse is that it’s not okay at all…”
Mr Hvid describes himself in his YouTube profile as an “urban explorer” and many of the images on his Instagram, Flickr and other social media feeds feature him at the top of buildings in places around the world.
Others pictures on his photography website, most of which feature women in unusual places, are labelled “nude”.