No. Every President could claim he could ignore the courts whenever he didn't get his way. You do not want to live in that country.
Separation of powers doesn't mean the President can ignore the court. It means the court decides what the law means, but if Congress and the President don't like it they can change the law.
Regarding this specific case, I've reviewed the decision quickly, and here's the deal.
Because this is an appeal the burden is on the government to show why the lower court should be overturned. It's only a restraining order, not the underlying case, so it's an easy decision to leave things alone. That's basically what they did.
The key issue is due process. Anyone in the country, even illegally, has due process rights. And due process is in the Constitution, neither Congress nor the President can ignore it. They found that the order didn't allow for due process for anyone here. BUT, they also acknowledged that this issue doesn't apply for aliens outside the country. They simply chose not to do a partial reversal, or in their words, rewrite the order. However I see no reason the President couldn't issue a new order, if he wanted to, that limited itself to aliens outside the country who do not already have visas. There is no due process issue there. This court has admitted that much.
What I find hilarious is that the kritarchists are now arguing that a nationwide TRO is just fine.
They argued the exact opposite when Judge Hanen issued his order in the DACA case brought by Texas:
http://www.scotusblog.com/2016/11/academic-highlight-bray-on-reforming-the-national-injunction/
There is nothing in the order regarding persons already present in the US.
Other than the Adminstrative parts, there are only a few "bans" in the whole order:
Section 3 (c) - "... pursuant to section 212(f) of the INA, 8 U.S.C. 1182(f), I hereby proclaim that the immigrant and nonimmigrant entry into the United States of aliens from countries referred to in section 217(a)(12) of the INA, 8 U.S.C. 1187(a)(12), would be detrimental to the interests of the United States, and I hereby suspend entry into the United States, as immigrants and nonimmigrants, of such persons for 90 days from the date of this order (excluding those foreign nationals traveling on diplomatic visas, North Atlantic Treaty Organization visas, C-2 visas for travel to the United Nations, and G-1, G-2, G-3, and G-4 visas)." Exceptions are specifically allowed for.
Section 5 (a) "The Secretary of State shall suspend the U.S. Refugee Admissions Program (USRAP) for 120 days."
Section 5 (c) "Pursuant to section 212(f) of the INA, 8 U.S.C. 1182(f), I hereby proclaim that the entry of nationals of Syria as refugees is detrimental to the interests of the United States and thus suspend any such entry until such time as I have determined that sufficient changes have been made to the USRAP to ensure that admission of Syrian refugees is consistent with the national interest."
Section 5 (d) "Pursuant to section 212(f) of the INA, 8 U.S.C. 1182(f), I hereby proclaim that the entry of more than 50,000 refugees in fiscal year 2017 would be detrimental to the interests of the United States, and thus suspend any such entry until such time as I determine that additional admissions would be in the national interest."
Section 5(e) actually allows for exceptions.
In each and every case it bars ENTRY. Anyone already IN the US is NOT covered under the EO - IOW, these people will not lose any "right" they already possess. If they leave the country and want to return, then they are subject to the new rules.