How would Sauron fare?

[Gwaihir the Windlord]

Sauron was an extremely powerful Maia. He was not the Lord of evil, but he was certainly a major supplementary figure. Melkor was the Vala, and so of course he was the greater, but Sauron was nonetheless a great Power.

He is described in the Valaquenta:

In all the deeds of Melkor the Morgoth upon Arda, in his vast works and in the deceits of his cunning, Sauron had a part, and was only less evil than his master in that for long he served another and not himself. But in after years he rose like a shadow of Morgoth and a ghost of his malice, and walked behind him on the same ruinous path down into the void.

He 'walked behind him'; a 'shadow'. Sauron could not have done what Morgoth did, i.e. building up the force of darkness in the world and weaving his evil into the very fabric of Ea, but once that work was done, Sauron was well able to take over the command and position that Melkor had previously filled. Sauron's spirit was what drove the hordes of Orcs that he commanded, and his mental power was very great. Gandalf remarks that it is greater than Saruman's and Denethor's also. It might be true that Melian may have been a match for Sauron mentally, but her power was not like his. She could take a little land and fence it out, and counter the actions of her enemy, but the force that Sauron commanded -- assisted by the works than Morgoth had wrought before him -- was greater than her.

Perhaps most importantly, Sauron was not simply a solitary power as he may appear to be. As I have said, Morgoth's power that still resided in the world was at Sauron's command; unlike Morgoth, who had to create it all and build it up, Sauron was able to take up and wield the weapon of fear and evil that Morgoth had already left him. Melkor had also not left the scene, even though he had been defeated and ousted from the world, as we know he was still able to exert his influence -- through his servants, eg Sauron, through his spirit and through what he had left behind on earth -- into Arda. Sauron, in this respect, was riding Morgoth's wave.
I think we can conclude that Sauron could not have done what Morgoth did to begin with, but that as a successor, he was equally fearful.