There's a fundamental misunderstanding in your question: the famous quote of Jackson's ("John Marshall has made his decision; now let him enforce it.") probably is apocryphal and there was nothing to enforce in Cherokee Nation v. Georgia. This is a historical sub and not a legal one, so I'll spare you the legal analysis, but what you need to know is that the Chief Justice Marshall, writing for the majority, wrote:

If it be true that the Cherokee Nation have rights, this is not the tribunal in which those rights are to be asserted. If it be true that wrongs have been inflicted, and that still greater are to be apprehended, this is not the tribunal which can redress the past or prevent the future.

This means the Court held that the plaintiffs lacked standing to bring the suit and that the Court had no power to redress the wrongs alleged by the plaintiffs. The case was not heard on its merits.

Perhaps the case you're thinking of is Worcester v. Georgia. But again, there was nothing to enforce. The Court ruled that the state of Georgia's conviction and sentence of Worcester was "void, as being repugnant to the Constitution, treaties, and laws of the United States, and ought, therefore, to be reversed and annulled."

Note that the federal government was not involved, and so Jackson didn't have to take any steps to enforce the judgment of the Supreme Court.

Being that the Executive is the only one with the power to execute laws, and in this case control the military, there isn't much else that can be done. Neither Congress nor the courts have the sort of enforcement powers that the Executive Branch has and so there's really no way that the courts can compel the Executive to do anything in the case that the Executive refuses to comply. What small enforcement they do have is the Marshal of the United States Supreme Court, which is in control of the Supreme Court Police. But these agencies, while independent and beholden to the Court itself, are not powerful enough to do anything about Executive action.

The relationship between the branches then is highly dependent on voluntary compliance, but also buttressed by the checks and balances designed in the Constitution. In some ways that makes it a shaky situation, as Jackson demonstrated. But all in all, respect for the rules of the game has led to a very stable relationship between the courts and the Executive. Further, the strength of and deferment to the Court has increased subsequent to Jackson's time, as can be evidenced e.g. by Nixon's complying with the result of US v. Nixon, and perhaps also by Gore's choosing to concede after the results of Bush v. Gore.

If you're thinking that there's room for exploitation here, you could very well be right. What would happen if someone with less than scrupulous morals won the presidency and decided not to defer to the Courts or to Congress? Well there are means by which the Executive Branch can be limited, for example by Congress refusing to fund them, impeachment in some circumstances, and hopefully the People would act as a check by refusing to vote for that person again, but the worst of circumstances are possible and a lot of damage could be done in a single term by the right person.