No, I'm afraid not. The Torah tells us to celebrate it on the first day of Tishrei, which is determined by the lunar calendar (and not the solar calendar, by which one determines the solstices). The lunar calendar, however, is indeed adjusted to more or less match the solar one (by adding leap months and days). Therefore Rosh Hashanah is always near the fall equinox (and not the winter solstice).

You may have been confused by the Gemara in Avodah Zarah which explains that the idolators had two 7-day holidays based on the winter solstice. The holidays stemmed from Adam, who noticed the days becoming shorter after his sin and thought that the world was coming to an end. When he saw the days begin becoming longer again he made a celebration. The idolators took up the celebration, but used it to praise their idols (Avodah Zarah 8a).

Be well,

M. Kornfeld

DAF