First, would the Supreme Court need the approval of the President (or the Executive Branch) to enforce its decision [in the Cherokee cases, when the U.S. Supreme Court decided in favor of the Cherokee, and President Jackson defied the court order and drove the Cherokee out of Georgia]? Second, has the Supreme Court ever actually tried to enforce a decision opposed or resisted by the President?The short answers to each question are yes, consent of executive officials is needed, and no, not in any important way. But the questions are important, and deserve more discussion.
First, this is not just an issue for the U.S. Supreme Court. In the United States, courts in general do not have direct line authority over armed enforcement agents. Federal courts once did, for U.S. marshals, under the Judiciary Act of 1789, but that authority was steadily eroded, and formally ended in 1969. Judges or court administrators might hire, fire, promote, or reassign clerks and bailiffs, but no longer most of the armed agents they might need to enforce their orders. However, that does not mean that as a matter of custom, policy, and practice individual agents seek the consent of their superiors before carrying out any court order, or that they might not defy their superiors to carry out such orders if there were a conflict.
Ultimately, all decisions by government officials depend for their enforcement on the voluntary assent of enforcement agents, from clerks to military personnel, or of the people generally. When conflicts develop between the will of line or judicial superiors, and the law, as perceived by each agent or individual, then each agent or individual has to decide for himself which to support.
In theory, every government agent takes an oath to follow the law, not the will of officials in conflict with the law. That requires that every agent, or for that matter, every individual, make an independent determination of what is the law, resolve any conflict of laws, and help enforce the law, regardless of what a judge, a superior, or a legal adviser might say. The argument in Marbury v. Madison applies not just to judges, but to everyone, in any legal issue in which one might become involved. Ultimately, having a written constitution of government means we are all on our own, required to each become an expert on constitutional and legal construction.
In practice very few individuals have the will or the skill to fulfill that awesome responsibility, and therein may lie the flaw in the very concept of constitutional government, that it requires more of most human beings than they can bear. But for those of us who have taken the oath to "preserve, protect, and defend the Constitution", and who take that duty and our honor seriously, the duty is ours, whether we can bear it or not, and we must do the best we can.
So, what would have happened if the Supreme Court had ordered U.S. marshals to enforce its orders in the Cherokee cases, in conflict with President Jackson and local officials in Georgia? They would have been outnumbered and outgunned, which is probably why that option was not pursued. The array of forces on the ground did not make it feasible.
However, historically and originally it was less paid law enforcement agents to which courts looked to carry out their orders, than to militia. Courts could and did call up militia for that purpose, and the people, as militia, if they respected the authority of the courts, as they usually did, would enforce the court orders. In the early republic, government was mostly local, and mostly carried out by juries and militia. See "The Jury and Consensus Government in Mid-Eighteenth-Century America", William E. Nelson, at http://constitution.org/jury/pj/nelson.htm . But that was for local courts and local judges. Had the U.S. Supreme Court tried to command local militia in Georgia to carry out its orders, it would likely have been ignored.
And that brings us to the key point: If people ignore the decisions of courts or other officials, those decisions become empty gestures. It all rests on voluntary consent, from one decision to another. We can speak of the consent that is expressed in constitutional conventions or elections, but the real consent that matters most is the habit and custom of obedience, and if that fails, governance itself fails.
If current government policies bring the collapse of the world economy, we may see this point manifested in ways most people can barely imagine.