The supreme court is the ultimate court of appeals in the United States - once you lose there, your only remedy is to change the law, or in some cases, change the constitution.

Since the supreme court interprets the law, the way you to change the ruling is to change the law. In a few recent cases, the supreme court has basically said (paraphrasing) "we don't like this ruling, but this is what the law says. If you want to fix it, talk to Congress, not us."

There have been some very notable instances where the Supreme Court has changed its mind or modified its prior rulings (for instance around civil rights), so while its the end of the road for this petitioner, it doesn't bar the law evolving in the future.

In the US the Supreme Court is the final arbiter of how the law is interpreted. So, off the top of my head, there are at least 2 ways the effects of a judgement can be reversed. 1) The law can be specifically amended by congress and 2) the Supreme Court can make a future ruling which is different than it's original.

Of course, there are intricacies here. Especially in case 2 where the weight giving to legal precedence makes it unlikely.

Yes, it's very strong legally. Congress or the states can try for a Constitutional Amendment (but those are few and far between, last one was in 1992), or the Supreme Court can later reverse itself partially or in full. An example of that would be Plessy v Ferguson in 1896, where the Supreme Court ruled segregation laws were constitutional, but later in the 50's (Brown v. Board of Education) they reversed it.

Those are the only options if they're ruling on the constitutionality of a certain law. If they're just deciding on whether or not an action violated any given law (as in this case), you could reverse the ruling by changing the law itself (i.e. through Congress).

The funny thing about this question is that the Court previously (in the 1970s) did rule in favor of this type of service, at the time known as community antenna TV. Congress then passed a new, revised copyright law that specifically outlawed this type of activity; it was under this law that the Court ruled against Aereo.

So that answers your question about what the next step could be, as well as why the Court ruled the way it did.

Appealing to congress to change the law in question is the only recourse to the losing party.

What exactly do you mean by a “mistaken” ruling?