The whole premise that whatever it does say about the subject would overrule principles like equality is inherently flawed.
But for the record the Bible doesn't even mention democracy, and pretty much assumes monarchy and authoritarianism as the default.
It also says, however, that Christians should accept governments run by non-Christians and not try to establish a theocracy.
I will not quibble here with anyone about the truth of the Bible, but taking it at its word there are only 2 forms of government with God's full support. In the Old Testament, Moses wrote down the laws God told him and when they came into the land of Israel the followed those laws more or less (less largely) with no central government for several centuries. This is ostensibly what God wanted for them. When they clamored for a king a prophet anointed Saul, but made very clear that it wasn't what God's favorite thing and that they were going to regret it and they did. They would have had no notion of democracy. Nobody was voting.
Democracy may have been known to any of the New Testament writers. Certainly Paul who was a Roman citizen. But I expect it looked a bit like a failed form of government at the time. I'm not sure how much voting was going on at the time. Augustus is Emperor at the beginning of the New Testament. Judea was a province without citizenship and the Roman governor shared power responsibilities with a puppet king and a religious council. The main political ambition of Judeas was apparently to get their own king, because of a prophecy pointing to a future time when Jesus is King.
I got so distracted and am not realizing that I proved the guy's point in name. the Bible certainly doesn't promote universal suffrage. But I am assuming this isn't what he meant, because he isn't advocating monarchy or constitutional anarchy either.