They are the first in a swarm of lawsuits filed by the Trump campaign as counting continues in the US election.
The Georgia case was filed on Wednesday after a Republican witness said that he did not know whether a pile of 53 ballots were received on time.
But Judge James Bass dismissed the case on Thursday morning.
“After listening to the evidence, I’m denying the request, dismissing the petition, thank you gentlemen,” he said at the end of the hour-long hearing in Savannah.
Trump holds a 12,000 vote lead in Georgia with about 47,000 votes still to be counted.
In Michigan, a judge has also rejected the Trump campaign’s lawsuit aiming to stop vote counting there.
The ABC has reported that Michigan Court of Claims Judge Cynthia Stephens made a ruling against the bid aimed at halting voting in the state. A written ruling is expected to be issued on Friday, US time.
Meanwhile, the Trump campaign has been able to enter the vote counting centre in Philadelphia thanks to a court ruling in their favour, the ABC reports.
A state court ordered a judge to ensure that party and candidate observers can get up close to election workers processing mail-in ballots in Philadelphia.
The decision came after the Trump campaign complained that its observer could not get close enough to election workers to see the writing on the mail-in ballot envelopes.
The specific complaint was made to ensure that the envelope contains a signature and an eligible voter’s name and address.
Ballots without that kind of information could be challenged or disqualified.
US voters are soon expecting updates in the handful of states still in play in the hope of getting a decisive result on whether President Donald Trump or Biden had won.
Biden has a lead in the race and has multiple pathways to victory while Trump is in a tighter spot.
All eyes will be on Nevada, Georgia and Pennsylvania to see if any are solid enough to be called for a candidate.
If Biden can hold a lead in Arizona, which is also set to report more results, any other state coming in for him will likely push his campaign over the line.
The election has been a nail-biter, with the Democrats having failed to deliver the repudiation of Trump that they had promised.
The Republicans look set to hold a slim majority in the Senate while the Democratic majority in the House of Representatives has shrunk.
Several of Trump’s posts on Twitter and Facebook have been affixed with warning messages, including ones claiming “fraud” without proof, amid concerns they were designed to spread misinformation.
President Donald Trump’s campaign says it expects to launch additional legal action in Pennsylvania and Nevada and predicts the Republican incumbent could emerge victorious in the US election by as early as Friday evening.
Trump campaign manager Bill Stepien told reporters on a conference call on Thursday that Trump was “alive and well” with regard to the presidential race.
Campaign adviser Jason Miller said he expected legal action in Pennsylvania to ensure visibility on previous ballots that have been counted in that state.
The Trump campaign also said it was confident the president would ultimately pull out a victory in Arizona, where votes were still being counted, including in Maricopa County, the state’s most populous area.
“All of the recent Biden claimed States will be legally challenged by us for Voter Fraud and State Election Fraud,” Trump wrote in a tweet on Thursday, without listing specific states or examples of alleged fraud.
– with AAP
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