And Christie argued last month that the de facto Republican leader was "afraid" of debating serious opponents.
Trump has indicated that he may skip at least one of the first two Republican primary debates, expressing a reluctance to share the limelight with lower-polling rivals.
"If he really cares about the country - and I have deep questions about that - but if he really cares about the country, then he's going to get up there, and he shouldn't be afraid," Christie told conservative radio host Hugh Hewitt.
Christie has assailed Trump on all manner of issues, highlighting the escalating criminal probes targeting the embattled billionaire, trashing his false claims of election fraud and dubbing him "Putin's puppet" over his isolationist stance on Russia's invasion of Ukraine.
John Ellis, a former head of the Fox News Election Night Decision Desk, told AFP that while a bigger field benefits Trump, Christie's candidacy could put the top-polling candidate on the back foot.
"Because he will attack Trump relentlessly, Christie's campaign will get a ton of coverage in the mainstream press, which may help him do well in New Hampshire," said Ellis.
Christie's launch comes a day after former vice president Mike Pence filed the paperwork for his own White House bid, setting up an unusual scenario in which two former running mates become rivals.
The evangelical Christian plans an official campaign launch on Wednesday in the early-voting state of Iowa - joining an already crowded field that includes Florida governor Ron DeSantis, Trump's closest competition.
Former governors Nikki Haley and Asa Hutchinson are also in the race, while North Dakota's chief executive Doug Burgam is due to announce on Wednesday.
Polls show Trump as the overwhelming early frontrunner, regularly posting leads on DeSantis in excess of 30 points. None of the other candidates is achieving double figures.
But the former president is engulfed by multiple criminal investigations into his personal and professional conduct that have already yielded dozens of felony charges in New York.
Legal watchers expect Trump to face further indictment in the coming days over the mishandling of government documents taken without permission from the White House and stashed improperly at his residence in Florida.
