Reuters is reporting the "suspects" could face up to 100 lashes in public.
"We are completing [the paperwork] and will soon hand over [the four people] to prosecutors," said Marzuki, head of sharia police investigations.
The action has been slammed as torture by Graeme Reid, director of the LGBT rights program at Human Rights Watch. "Acehnese authorities should release the four and protect the public from marauding vigilantes."
Secular Indonesia is predominantly Muslim however ultra-conservative Aceh is the only province to follow sharia (Islamic) law which criminalises gay sex.
Last year, Vice reported on an almost identical vigilante action against gay sex which resulted in punishment of public lashings (video).