Quite often, a Supreme Court ruling decides a major point of constitutional meaning and rights. That's more powerful than a mere law.

Consider but one famous example, Brown v. Board of Education in 1954. That decision overturned the earlier "Separate but Equal" ruling allowing for segregated schools. When the Brown decision came down in 1954, it was quite controversial. Three years later, Arkansas Governor Orval Faubus (Democrat) openly defied the law. President Eisenhower (Republican) ordered in soldiers from the 101st Airborne Division to uphold the law. Later presidents, both Republican and Democrat, have used federal law enforcement officers to enforce Supreme Court decisions like Brown. I don't know of a single example where any president has used an executive order to go against a court decision.

Consider a highly controversial decision, 1972's "Row v. Wade" decision. Has any Republican president issued an Executive Order overturning the decision?

Here's a good overview on Executive Orders that explains their constitutional foundation. They can be challenged by Congress or in the courts. Not too many have been overturned, but there are means to do so.