Does a people have the right to claim a God-granted right against a government (established by God, but run by men) that denies them of it?
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By right I mean a moral entitlement.
Sure. But the Christian does not have the right to use violence to achieve that.
For example, the greatest right we have from God is the right to worship him as Christians. If the government decided that Christianity was illegal then I have a God given right to completely ignore what this government says and worship God anyway (Acts 5:29). However, that does not give me the right to use violence to acquire the legal permission of the government to worship. I would also say that the example of the apostles and the early church
indicates that even in this situation, breaking an unjust law in order to obey God, we ought to submit to the punishment of the government if caught. Of course the same examples show that we should avoid being caught and escape if possible.
Applying this to self-governance I would see no problem with working within the current system in order to establish a republic. I have a hard time seeing self-governing as a 'moral entitlement' from God as you put it, but I freely admit I haven't really spent much time thinking about it. If it is such a thing, and it is denied by the current government (say it's a completely totalitarian regime) the Christian still does not have the right to use violence in order to establish self-governance.
Is a government, borne of rebellion against a regime that denies the people their right (of self-governance), legitimate?
Good question. Essentially you're asking if the US is a legitimate government. However I think the problem here is that we are throwing in categories that the Bible never distinguishes or recognizes (to my knowledge). When Jesus, Peter, and Paul speak about submission to the governing authorities I do not see any caveat regarding the "legitimacy" of said authorities.
In fact I think you could argue that the Roman government they were under could be questioned on the same grounds, since the empire was established essentially by rebellion against the preceding Roman Republic (depending on what event you use to determine the end of the republic).
So I don't think "legitimacy" is really a question that scripture worries about, instead we are to submit to the authorities in power.
Let me ask you something. If self-governance is a moral entitlement that all people have, then were the apostles and the early church morally wrong to not seek to change the Roman government?