There is certainly a strong correlation between democracy and GDP/capita—however, it is just that, a correlation. There is no requirement—if there was, India wouldn't have sustained a democratic polity since 1947 (current PPP GDP/capita, approximately $2,700).
Indeed, quantitative studies that test correlates of democracy typically code India as a single case, on a par with, say, Vanuatu. If you weight it by population (1 billion), then the income/democracy relationship is far less clear.
More to the point, it is worth considering the case of Mozambique: despite poverty and 17 years of brutal civil war (almost 1 million dead, around 5% of the population), it has sustained imperfect democratic politics since the 1994 elections.
(Full credit to the UN, and more particularly the then SRSG Aldo Ajello, for managing a very tricky transition.)
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