Does the POTUS have unilateral authority to order a nuclear launch?

Yes.

Does anyone have the authority to stop him?

No.

A resource thread ⬇️ 1/x
How would a Presidential nuclear order go down?

@derekjGZ breaks it down step by step:
6/x https://twitter.com/derekjGZ/status/1347583047189860352?s=20
Need official government proof that the POTUS has unilateral authority to launch nuclear weapons and no one can stop him?

Here you go, from December 2020:
7/x
https://crsreports.congress.gov/product/pdf/IF/IF10521/7
Could the military refuse an illegal POTUS nuclear order?

Under certain circumstances, yes; but what constitutes as "illegal" is much more limited that you might imagine:
11/x http://blog.nuclearsecrecy.com/2017/04/10/president-bomb-iii/
Nuclear experts @ErnestMoniz and Sam Nunn present a legal analysis of the authorities, limits, and process of Presidential nuclear authority:
12/x
https://www.nti.org/newsroom/news/nti-releases-new-legal-analysis-president-and-nuclear-weapons-authorities-limits-and-process/
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