Whether a ‘Made in Europe sounds like the sort of question to which there ought to be an easy answer. It is clear from the UK cosmetic regulations that you need to use the country name. But it turned out not to be something that the regulations are clear on. This may be because the relevant US regulation, the Tariff Act, goes back to 1930 before anyone had even thought about the European Union.
Digging around my collection of official documents and some fairly intense Googling eventually turned up the answer. In a position paper about country of origin rules, the statement is made that the US does not accept the EU as a valid country of origin. The link it gives no longer works. But with that clue, I was able to track down the most recent guidance which dates from 2020 and confirms what the EU document says.
https://www.cbp.gov/sites/default/files/assets/documents/2020-Jul/ICPMarking-of-COO-onUS-Imports.pdf
