It is hard to imagine that Paul himself referred to the American Revolution or any modern event. However modern readers may certainly interpret his writing as applicable, or even prophetic. On the other hand, this teaching needs to be understood in historical context and balanced against other Biblical instruction which supports following one's conscience rather than obeying the ruling power. Here is the relevant passage in the RSV:
Let every person be subject to the governing authorities. For there
is no authority except from God, and those that exist have been
instituted by God. 2 Therefore he who resists the authorities resists
what God has appointed, and those who resist will incur judgment. 3
For rulers are not a terror to good conduct, but to bad.
Here, Paul is writing to the Roman Church, probably just a few years before the Neroan persecution of Roman Christians. Indeed he mentions "persecution" and "tribulation" as two of his specific concerns. (8:34-36) Clearly he wants the church to give the authorities no pretext to use against it. He would also be aware of growing tensions between the Roman Empire and the Jews; and Christianity was still seen as merely a sect of Judaism at the time. Thus, he rightly counseled Christians not the rock the Roman boat, so to speak.
Looking at other biblical text we can find teachings that both support and disagree with Paul's attitude here, including from Paul himself:
The teaching to "render unto Caesar" (Matthew 22:21) is an example of agreement with Paul's policy: Jesus teaches Jews to pay both the Roman tax (with Roman denarii) and the Jewish temple tax (with Jewish shekels). The context is a situation in which Zealots and a certain faction of the Pharisees were counseling resistance to Roman authority.
Jesus' overturning the moneychangers' table is an example of his acting against Paul's later advice, because the moneychangers were an essential part of the Temple system approved by both Roman and Jewish ruling authorities.
When Peter and other apostles ran afoul of authorities in Jerusalem, the ruler said: "We strictly charged you not to teach in this name, yet here you have filled Jerusalem with your teaching and you intend to bring this man’s blood upon us.” 29 But Peter and the apostles answered, “We must obey God rather than men." (Acts 5:28-29)
Paul himself "boasts" of being punished by both Jewish authorities (39 lashes on five occasions) and Roman magistrates (beaten with rods) during his travels. (2 Cor. 11:24-25; Acts 16:20)
There are many other examples that may be cited on both sides of the "conscience vs. obedience" ledger.
Regarding the Revolutionary War, recent research into 'The Bible and the American Revolution' shows that Roman 13 was often quoted in Anglican sermons prior to the Revolutionary War. Preachers who supported Independence tended to quote the Song of Deborah (which praises a revolt against Philistine rulers) or Exodus (Moses delivering the Israelites from bondage).
How Paul would have felt about the American Revolution is anyone's guess. Both England and its colonies were predominantly Christian, so he need not worry about the church's very existence, as he did in the case of the church at Rome. There, he urged the Roman Christians to avoid running afoul of authorities, but in other cities he was more than willing to speak truth power, and suffered the consequence proudly.