The clip posted by Trump's daughter Ivanka shows 5-year-old Arabella Kushner singing best wishes for the holiday while playing with a traditional Chinese puppet.
On Wednesday, Ivanka and Arabella made a surprise visit to the Chinese Embassy in Washington to take part in the festivities. Arabella had already drawn considerable attention in China in November when a video clip of her reciting an ancient Chinese poem lit up the internet.
While the Trump women received praise, the lack of a greeting from the president "triggered heated discussion" among the Chinese public over the possible ramifications for bilateral relations, the official newspaper Global Times said Friday.
The paper noted that Trump's most recent predecessors Bill Clinton, George W. Bush and Barack Obama "all delivered Lunar New Year greetings to Asians who celebrate this festival.
"Some Chinese (internet) users felt Trump was being disrespectful by not sending a greeting," the paper said.
The Lunar New Year, which this year fell on last Saturday, is China's most important traditional holiday, a time for family gatherings, feasts and the exchange of greetings between relatives, friends and work colleagues. It is also celebrated in several other Asian countries and among Chinese communities worldwide.
Comments on the internet largely focused on what many Chinese view as a hawkish attitude by Trump and his administration toward China, with many saying the lack of a greeting should be seen as a deliberate snub.
Trump has accused Beijing of manipulating the value of its currency to gain an advantage in trade and threatened to impose a 45 percent tariff on imports from China. He's also criticized China's military buildup in the South China Sea and accused Beijing of doing too little to pressure neighbor North Korea over its nuclear and missile programs.