Americans woke up on the last day of 2016 to a message from President-elect Donald J. Trump wishing them a happy new year.

But the holiday missive, posted on Mr. Trump’s official Twitter account around 8 a.m., came with a pointed jab.

“Happy New Year to all,” Mr. Trump wrote, “including to my many enemies and those who have fought me and lost so badly they just don’t know what to do. Love!” (Some social media users homed in on Mr. Trump’s use of the word enemies, rather than opponents or another word for those who oppose him.)

Even a holiday known for its good cheer, it seems, could not dull Mr. Trump’s penchant for doling out taunts and insults via Twitter.