By itself, the energy contained in a plasma (which by definition includes electrons and ionized atoms) is not enough to cause fission or fusion. However, if you add energy to the plasma, such as by heating it, you can get nuclei to fuse together. This is how magnetic-confinement (Tokamak) fusion reactors work, such as the proposed ITER reactor that the global fusion community wants to build. In this case the plasma is not a "requirement", but rather a "result" of the fact that you need to heat your fuel to many millions of degrees -- at which point a completely ionized plasma is all that exists.
On a related note, you can get atoms to split by hitting them with other atoms, but again you generally need to add energy to the reaction, such as with an accelerator. Alternatively, a nuclear reactor can do it without adding energy, by taking advantage of the neutral charge of neutrons (no energy is needed to get them close to other atoms), and the relatively large inherent energy they bring to a nucleus that absorbs them.
In short, whether or not a material is ionized plays no part in the process of fusing or fissioning atoms. Ionization involves energy of the electrons (typically in the electron-Volt range), while the breaking up or fusing of nuclei involves energy of the nucleus itself, which is millions of times greater.
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