I think he'd win, especially if you just crouched. If you jump around and over him like a mad man, though, it'll be a lot harder for him to aim at you, a twichy-moving target, than you to hit him, a near stationary target.
It's not too hard to kill Heavies as a Scout you just HAVE to use your speed to your advantage. If he has a Medic, though, I wouldn't even bother fighting him. I'd try to put some hurt in the Medic and leave before I get torn to shreds by Sasha, or Natascha.
For medics... don't get too close if they can see you and just shoot away. I like to use my first shot point blank, second one backing, maybe a third if he isn't dead and pistol him if he still isn't. Overall, it's easy if they don't have backup, but if they do(which they should)... it can be a challenge.
@GMan: I don't know why, I find that the Bat is a really good melee weapon, and I've gotten a lot of kills with it. I don't even always use it as a backup weapon, just in the right situations. Sure a Scattergun shot is often more damaging, but because of the Scouts speed and rate of swinging, it's pretty easy to run, jump and dodge all while meleeing fast, and hopefully critting(assuming crits are on, otherwise melee weapons aren't all that useful anyway.) It's generally easier on slower classes, even with their high firepower. I've killed many a Soldier, Demo, etc with the bat, it just takes some practice. The only classes this doesn't work on for me is Pyro and good Heavies. But maybe I'm just a good batter, I've always been good with melee weapons.