Yes, the United States will pressure its allies in the Persian Gulf to respect habeus corpus, because this is something that we currently do. We have pressured Kuwait. We have pressured Iraq in two separate wars in only a generation. We always want other nations to adopt our way of living.
It is possible that the United States could use this as a bargaining tool for something they want from our allies. However, why would other countries, that follow their own system, decide to change to our system? Fairness may not be a concern with criminals in those these countries. However, if it impacts a US citizen, then I think the United States should attempt to intervene.
Persians have a different judicial system than people in the U.S. So the way they punished those accused of a crime or seen committing a crime could lead to absolute death. This is something that the United States is working on but Persian countries don't card anything about the United States and their beliefs.
I have my doubts the United States will pressure its allies in the Persian Gulf to respect habeus corpus. The United States doesn't follow habeus corpus in a lot of cases right now, so it would be hypocritical to suggest other countries should follow a precedent we can't even maintain.