Indeed, if a nucleus only has protons, it will be unstable, and split up, simply because protons all carry positive charges (same charges repel each other due to the Coulomb interaction ). Neutrons can stabilize a nucleus, because they roughly act like bonds between protons. They will bind protons together through what is called the strong force. The strong force is a very short range force which only dominates within very small distance, like the size of a nucleus.

However, the smallest nucleus, the hydrogen nucleus only has one proton, so it does not need neutron to be stabilized.

An atom’s nucleus is made of protons and neutrons. Protons are positively charged, and if you have ever played with magnets you know that positive charges like to be next to negative charges and do not like to be next to other positive charges. Scientists would say that “like charges repel each other.” So how does the nucleus stay together if all those positively charged protons do not want to be next to each other? Well, there is a force called the strong force which holds protons and neutrons together, which is much stronger than the repulsion between like charges. So because the nucleus has both protons and neutrons, the strong force can hold all of them together.

Without neutrons, the protons in a nucleus would fly apart since there would only be repulsion forces due to the positive charges not wanting to be near each other.

In science research, we have so far not seen nuclei with only protons, except for hydrogen. The most likely reason that we don't see other atoms forming nuclei with protons with no neutrons or electrons is that such a thing would not be stable. In other words, there are forces that would not allow such a thing to exist under normal conditions. A hydrogen nucleus with a single proton and no neutrons or electrons occurs when a hydrogen atom loses an electron, although this form does not exist for long because it reacts very quickly with things around it. This form is just a single proton, and it has a charge of positive one. Aside from this form of a hydrogen ion, aka a single proton, nuclei with protons only (no neutrons or electrons) do not commonly exist.

One answer is that an atom can be stable with a nucleus comprising only protons, but only in the case that the nucleus contains only a single proton. This is the case for hydrogen atoms (at least, most hydrogen - deuterium, tritium, and other isotopes, all with one proton and 1+ neutrons, also exist.)

Presumably you want more than a single proton in the nucleus though. A nucleus of many protons and no neutrons would not be stable because protons have a positive electric charge. Since charges of the same sign repel, protons repel each other (even in real materials which have neutrons in the nucleus). With nothing else to hold the protons together, this repulsion would rip the nucleus apart. As explained here, neutrons hold the nucleus together through the strong nuclear force.

I'm not sure what is meant by "Would it produce charge?". A nucleus of only protons would contain only particles with positive charges, so such a nucleus would also have a positive charge.

In addition to the electromagnetic repulsion, a nucleus of only (multiple) protons would almost certainly be radioactively unstable though, which seems to be the intent of the final question. Basically, the all-proton nucleus would be very high in energy, and to get rid of some energy it would undergo a nuclear process. While there are more types possible in real materials, the only ones available to this nucleus would be positron emission (releasing a sort of positively-charged electron from a proton) and electron capture (adding a negatively-charged electron to a proton). Both of these would reduce the nuclear charge (and number of protons in the nucleus) by 1 while increasing the number of neutrons in the nucleus.

The force that holds the atomic nucleus together is known as the strong force. Recall that protons are positively charged and repel each other by the electromagnetic force (a positive charge repels another positive charge). The reason that the positive nucleus doesn’t fly apart is because of the strong nuclear force which acts between protons and neutrons and “glues” them together. This force is caused by the exchange of a particle known as a ( gluon ), acting between quarks (which are the subatomic particles that make up both protons and neutrons). Removing neutrons from a nucleus would decrease the strong force while increasing the electrostatic repulsion of protons (by moving them closer together). Eventually, if enough neutrons were removed, the electrostatic repulsion might become stronger than the strong force holding the nucleus together and the nucleus would break apart.

The most common isotope in the universe is hydrogen-1, which consists of a single proton and no neutrons. Approximately 90% of the atoms in the universe are of this isotope.

Larger atoms with more protons will undergo inverse beta decay, emitting a positron and converting one of the protons into a neutron with each decay. This will cause an unstable proton-heavy isotope to reduce its number of protons and pick up neutrons until it becomes stable.