"God, from all eternity, did, by the most wise and holy counsel of his own will, freely, and unchangeably ordain whatsoever comes to pass: yet so, as thereby neither is God the author of sin, nor is violence offered to the will of the creatures; nor is the liberty or contingency of second causes taken away, but rather established." —Westminster Confession of Faith, chapter 3: 1.
If, as I believe is true, and as makes sense by way of causation, that God, being the uncaused first cause, causes all things subsequent to himself, then the scripture is affirmed, not only that Government is intended for the people's good, but that God installs and even directs the rulers, and that they are to be obeyed.
In this, as you mentioned concerning Saul and David, (and many others, btw), that the respect is not necessarily implied between one king and another —that is, between one country and another— the recognition that they are installed by God is. It would be implied, then, that serious reasons and Godly motivation would be required to take down the ruler of an opposing country. But to depose by violence one's own ruler (of one's own country) requires direct edict from God.
But that even that is done sometimes is evidence that God caused it, by means of (usually) disobedience, and that, not necessarily because of the worth of the ruler deposed, but because God causes all things to work together for good, for them to whom he chose to show mercy —i.e. the Bride of Christ. This is the reason, for his own glory, that God created, and it doesn't have to look right to us.
This coming [supposed] "election", whether by fraud or not, a new President is to be chosen. That will fall out however it does so, according to the decree of the King of Kings.