The complaint was filed less than 24 hours before Alabama election officials certified Jones as the victor, paving the way for his swearing-in to the Senate next month.
"It's a very weak complaint, it does not rely on any election experts I've ever heard of and it makes claims that on their face do not appear to be plausible, or appear to be irrelevant", said Rick Hasen, an election law professor at the University of California, Irvine. Jones said he will be an "independent voice" in the Senate and will strive for "common ground with my colleagues on both sides of the aisle".
Jones is the first Democrat in a generation to win a Senate seat in Alabama, beating Moore amid a firestorm of allegations that the GOP candidate had sexually abused teens.
Moore appeared to be the favorite in the contest to fill the Senate seat held by Jeff Sessions, who is now Trump's attorney general.
The certification was complete at 1:10 p.m. Thursday.
Moore was defeated in the race by Democrat Doug Jones. A judge denied his request to stop the election certification.
Merrill said the allegations of voting irregularities by Moore's team had been thoroughly investigated and found to be baseless, while a judge threw out a suit seeking to delay the results certification.
Sam Coleman, a spokesman for Jones's campaign, was not impressed by Moore's complaint. He said he had no regrets.
"The people of Alabama have spoken and Moore lost". Jones had a lead of 20,715 in the unofficial returns and was ahead 21,924 in the certified result.
Snow pushes out; arctic air coming soon
Very cold air looks to move into the region next week with highs in the mid to upper 20s and lows in the teens and single digits. Temperatures have to moderate a bit next week, and we may get a better shot at some snow or a wintry mix by about Thursday .
Photo Roy Moore addressing supporters at a rally in Midland City, Ala., a day before the December 12 Senate election.
Jones is a former US attorney best known for prosecuting two Ku Klux Klansmen responsible for Birmingham's infamous 1963 church bombing.
Mr. Merrill did not immediately respond to a text message.
Janet Porter, a spokeswoman for Moore's campaign, told CNN in an interview on Thursday that the challenge aimed to ensure that votes were properly counted. Once was for defying a federal court order to remove a Ten Commandments monument from the state judicial building. Moore's campaign was deeply wounded by accusations of sexual misconduct involving teenage girls decades ago.
Mr Moore denied the claims and said he has taken and passed a polygraph test to prove they are false.
Moore's complaint cites four so-called experts on why there should be a new election, but some of them come with "baggage", the Post reports. He attached a statement from a poll worker that she had noticed licenses from Georgia and North Carolina as people signed in to vote.
Moore's complaint also alleged "anomalous" higher voter turnout in Jefferson County, in which census data shows 43% of the population is black.
"This is not a Republican or Democrat issue as election integrity should matter to everyone", Moore said in a statement Wednesday.