The reason why Na+ and K+ are very unreactive is that while they are positively charged, they are well insulated from any chemical reactions by the large and very stable electron clouds of neon and argon octets. While the electrostatic attraction of the positive charge will attract counterions and dipoles in solution, they are unable to get close to sodium nucleus - several of them will form a diffuse surrounding, and none of them can form either a covalent bond or give away an electron to the sodium ion. Even if you do force sodium out of solution, it is still Na+ cation in ionic cristal, without covalent bonds to the counteranions, and if you force electron into sodium atom, it is weakly bonded and readily given away to metallic electron sea, wher sodium is still Na+ ions.
Compared to Na+, H+ is a small and concentrated charge, and H can easily form covalent bonds. This is what makes H+ a strong acid - its tendency to form covalent bonds to electron donors.
There are other things which have the same tendency to form covalent bonds to electron donors. For example, consider Al3+. AlCl3 is even electrically neutral. Yet Al, with its empty p orbital and a partial positive charge, has a strong tendency to form covalent bonds to electron donors. It is actually called acid - with qualification that it is a Lewis acid.
Al3+ can, among other electron donors, react with water. Thus aluminum salts react with water and make the solution acid in containing H+ ions. The reaction is
Al3+ + H2O->AlOH2+ + H+
Note that although Al is a strong electrophile/Lewis acid, it is not an oxidizer. Al has a strong tendency to react with compounds having an available electron pair for a covalent bond - but no tendency to add an electron and not form a bond. AlOH2+ and other such adducts are stable, but Al2+ or Al+ ions are very unstable.