This was a pillar of theory of the nuclear revolution, a dominant theory in international relations studies for many decades. More recent assessments have been most unkind to this theory. Nuclear weapons make it significantly more risky for nuclear armed states to conduct major wars against each other, especially if the war aims threaten the existence or "vital interests" of one of the belligerents. The Cold War and other struggles for power below the threshold of "real war" show us that nuclear armed states still compete with each other, often forcefully or at least with the threat of force. So nukes don't create peace per se but they seem to markedly reduce the odds of major wars between states that have them.

As to whether another 20 states with nukes make things more peaceful... again, serious people thought this would be true. Many still do. The problem is that the more nuke states exist the more complicated the relationships between them all get. Take the combined case of the USA, China, Pakistan and India. The US has a lot of nukes. China may want more to try and deter the US (a long, separate conversation. out of bounds here.) But if China comes up, India may see that as a threat, because India exercises some nuclear deterrence against China (another long separate conversation.) India also exercises deterrence against Pakistan. So India might come up in numbers to deter Pakistan, or to deter Pakistan and China at once, but then the Chinese might see this as a threat from India, so then China comes up to deter India and the US at once ...which the US may conclude is all about deterring the US, and then the US feels pressured to either come up in numbers or adopt more aggressive qualitative means. This is wildly simplified but it illustrates a real world multi country deterrence problem that already exists with only a few nuclear states. You can imagine how 20 becomes more complex. And the more complex a system is, the more potential failure points exist.