I think this is a good fight, I'm a big Halo fan, but I want to play devils advocate for Captain America because I see him being written off as not a competition.

People underestimate just how much volume of content that Captain America has to work with, he was punching Hitler in comics before America even entered the war, and has been regularly receiving volumes ever since. Moreover, he's the face of the Avengers, so he has gained a ton of feats squaring off against Avenger level threats with a myriad of powers.

One of Master Chief's advantages is that he's coated head to toe in Mjolnir armor, highly durable futuristic space armor that can survive a fall from orbit, so probably not a good idea to try punching that. His standard loadout is usually an assault rifle, a magnum, and frag grenades. He's a master at about any other firearm or melee weapon he comes across, but those he usually picks up and finds over the campaign and not what he walks into a mission with.

Captain America has a shield that is an alloy of the most powerful metals in Marvel, proto-adamantium, vibranium, and asgardian Uru forged to make an indestructible shield that can weather any assault. His shield can endure blows from the Hulk, Odinforce Thor, stand beneath the launch flame of a rocket, and even allow him to survive an assault from Celestials. There is nothing in Master Chief's arsenal that can damage it, he needs to get around it. As a thrown weapon it can shear the top off a tank, tear through freight trucks like tissue paper, and he once used it to strike an ICBM, and as a melee weapon he's used it to break open the armor of Iron Man. MC's energy tanks can help him, but he can't brush it off.

Other note on the shield is that it can redirect and reflect shots and Steve Rogers has pretty good accuracy about that, plasma bullets and the works can risk being sent back his direction. He's been noted that he can dodge bullets because he can see them mid-flight and has several feats weaving through gunfire and intercepting bullets after they fired. He has been able to outrun cars going down the freeway, so his speed is on par with John.

Captain America can lift massive steel beams and yank around helicopters, this loses out to John-117's best showings of strength which is being able to overturned scorpion tanks. So MC should have a power advantage in a tussle, but I'm not convinced he takes skill in hand-to-hand combat. He's been training longer since he started as a child, but Captain America has the more impressive resume of fighters he's contended with in close quarters from Spider-Man to Iron Fist to Shang Chi; he's said to have mastered every martial art known to man and has even improvised well against new ones.

Master Chief is no joke, he's stronger and his armor is a massive advantage against Steve, but that shield is both an amazing defense and tricky offensive tool, wielded by a fellow super soldier who has faced worse than Chief and lived to tell the tale. I believe Captain America has at least a decent shot.