Metroid's protagonist against the star of Vanquish in two specific rounds:
Round 1: Pure melee fight, wearing only their respective suits and with the basic capabilities of said suits (as in, no power-ups on Samus' suit, and Sam's suit has its basic limiters).
Round 2: All-out fight. Sam's got all guns available in the game, DLC guns included, and Samus has her fully-amped power suit, and Sam's ARS limiters are fully disabled.
Since Vanquish was a bit understated (the curse that lots of Shinji Mikami's most original games tend to have), if you don't know who the hell Sam Gideon is, here're a few random vids for an idea of what he can do.
For the melee battle I'd give Gideon the edge, if only for experience, since even after Other M Samus doesn't do a whole lot of close combat. Her style mostly consists of grappling to fire at point blank range, or a throw. They seem similar in physical strength, and Gideon looks a little stronger, but Samus is close enough that it wouldn't matter much, and she's used to fighting things stronger than her.
In the gun battle, I'm leaning farther towards Samus. Is there anything special about Gideon's bullets? Otherwise, bullets don't tend to do much to Samus. She's got weapons designed for overpenetration, and Gideon looks like he has trouble against heavy weapons. A few missiles or a super missile might wreck him. Gideon's fast, but I agree with Spy on the Ice beam point. Can he take freezing? It'd likely slow him down or stop him completely.
What else can Gideon do? I only saw strength, speed, and bullets/rockets in those videos. Does he have any particularly unusual guns? Defenses?
^Well, the most notable aspect of his suit is its ability called Augmented Reality mode, where everything goes basically into bullet-time after you dodge, are in the middle of a certain motion (like jumping over cover), or are about to die, and Sam himself is slowed down less than everything else. With the limiters, it runs on a meter.
As for his weapons, his guns are actually this sort of scanning weapon that can copy and become an exact replica of any properly-sized gun, and normally can only store three such scans at a time.
So he's got a time slow. Pretty cool, Samus has never faced anything like that before. The only thing even slightly similar is when she fights under super gravity and gets slowed down. Gideon's definitely a bit better in hand to hand, but Samus excels in grapples, so Samus' best option would be to not let him close or try to get a hold on him and throw him at something.
Everything from about 0:27 on is Samus' approach to melee. She seems more focused on counters and grabs than punching, but her kicks send things flying.
I still think Samus has the edge in ranged combat, though. She has infinite ammo, huge area covering shots, and some are designed to penetrate. Then she has missiles and super missile.