No, because there is no mathematical proof for
any assertion in economics. You cannot prove economics true.
However, historical counter-examples of zero-growth democracies are limited. We might look to the Greeks or Romans as examples of democracy in the antiquities; but there's two problems with that:
1. Their democracies were limited, with no universal suffrage and very low citizenship rates.
2. They had economic growth anyway. Rome in particular was as addicted to military conquest then as America is to oil today.
Selectorate theory[0] would argue this point in the opposite direction; i.e. that strict dictatorships squander economic growth. Political and economic franchise are inherently linked: people who cannot vote will have their livelihoods taken away from them, while those who produce the economy's goods will eventually demand political power themselves. Democracies are ultimately the means by which we distribute political power among the people, and that is itself a reflection of the distribution of economic production. If you give the people political power, the economy can't help but to grow as they are no longer being shackled.
But I think there's an even simpler argument than that. If the economy is zero-growth, then the only way for an individual to improve their living conditions is to make someone else's worse. This turns the economy into a zero-sum game of who can monopolize everything the fastest, which spikes income inequality, which makes it easier to disenfranchise voters, and eventually results in the ossification of social and political structures (since you shut up everyone that you said didn't matter).
[0] Excellently summarized by CGP Grey here: https://www.cgpgrey.com/blog/rules-for-rulers based on a book called The Dictator's Handbook