Not really true as a definitive statement. Beards came and went in both societies over the roughly 1000 year period we think of as "classical."
Both societies originally were bearded, of course.
Alexander the Great made shaving popular among Greeks, Scipio Africanus among Romans. Hadrian brought beards back in Rome, but they went out again with Constantine till well after Justinian.
The moral of the story being that it's really difficult to make broad statements about societies over a period of 1000 years plus.
Let’s hope they can say that about US history.