The 60m-tall structure was found underwater between the islands of Sao Miguel and Tercira. Its base measures 8,000 sq m - bigger than a football field - and Silva believes it was man-made.
He says his find may be a remnant of the legendary lost world of Atlantis, which sank into the ocean after a failed attempt to invade Athens.
Silva found the pyramid around five months ago while fishing on his yacht in the area. He discovered the structure through bathymetric navigation and its tip is submerged around 40ft.
"It's amazing because it forms a perfect pyramid. And moreover, orientation, deployment of the pyramid: the vertices are oriented north and south, just north and south, such as the Giza pyramids in Egypt," he told Portuguese news website Terra.
However, the Department of Oceanography and Fisheries at the University of Azores is less convinced of its mythical nature.
It is now being investigated by the Portuguese Navy, which is analysing the data underlying the discovery.
Admiral Fernando Pires, commander of the Maritime Zone of Azores, said there is not enough information about the pyramid at this time to say what exactly it is.
However, he said the structure does not pose a threat: "At the time we seek probes that could constitute a danger to navigation and found nothing," he told Sapo.
Pires said the navy has not ruled out the idea that the pyramid could have been formed by a volcanic eruption - the archipelago of the Azores is composed of volcanic islands.