As someone who really likes 5E... This seems like a misguided attempt to shoehorn in something from another system.
Fighters could just get an ability where they add +X to their weapon damage rolls-say, one third Fighter levels (rounding down).
They could also just get Second Wind-once per day at level one, they can regain 1d10+Fighter level HP. Get another use at level 5, then another at 10, 15, and 20.
I also would NOT add "Can only use on one weapon," without also adding something like the Warblade's Weapon Aptitude.