The charge of the proton does not transfer to the neutron because protons are charged particles.
For example, when we put two objects of +10C and +20C together and then take them apart, each of them acquires a charge of +15C.
This occurs because 5C's net worth of electrons (which are also charged particles – though with the opposite charge to protons) move from the +10C object to the +20C object. (And it doesn't always happen; only if there's enough electrical conductivity between the objects.)
This doesn't happen in the case of protons and neutrons, because protons are the charged particles. To transfer only some of the charge, you'd have to take the proton apart… but then you wouldn't have a proton any more. (And you wouldn't be able to get a fractional charge even if you did take the proton apart. There aren't any stable particles with a charge that isn't a multiple of an electron's charge – if you count “zero”, anyway.) Besides, taking a proton apart takes a lot more energy than is available in your situation, so it won't happen. so it very rarely happens; see Xerxes's answer.
Charge transfers because of charged particles moving around – or perhaps charged particles move around because the charge does. Any way you look at it, there is a smallest charge you can have, and there's no way to split it up any further than that, unless we discover new particles.