The diproton is in fact unstable. The two protons must have anti-aligned spins and the spin-dependence of the strong force leaves it unbound. In order for fusion to occur, the protons must get close together and one of them must undergo a weak interaction and change into a neutron (and form a bound deuteron with aligned proton and neutron spins). It is this combination that is very unlikely.
The Coulomb repulsion alone is not a particularly high barrier to fusion. For example, the fusion of two deuterons will occur at temperatures an order of magnitude lower than the fusion of protons inside a star, despite identical Coulomb repulsion.