The animal has twice tested positive for bovine tuberculosis and, as a result, the Department for Environment, Food and Rural Affairs (Defra) has ordered its destruction.
His owner Helen Macdonald, who imported him from New Zealand, believes the tests are returning false positives, but has been refused permission to have him tested a third time.
Ms Macdonald, who owns a farm at Wickwar, South Gloucestershire, has received an outpouring of support from the public, with more than 130,000 people signing a petition calling on Boris Johnson to halt the killing.
On Tuesday, an urgent application for a temporary injunction to halt the enforcement of the destruction order was considered by Mrs Justice Stacey at the High Court in London.
However, the judge said she would need further information from Ms Macdonald and from Government lawyers before she could make her decision, and said she would resume the hearing on Wednesday afternoon.
At court on Wednesday, the judge refused the urgent application and concluded there was "no prospect" of Ms Macdonald succeeding in her bid to reopen a previous ruling.
Ned Westaway, representing the Defra executive agency the Animal and Plant Health Agency (Apha), told the court the agency will not seek to execute the warrant on Wednesday evening, and would give Ms Macdonald the opportunity to make her own arrangements for Geronimo's destruction.
Ms Macdonald's lawyers told the court Geronimo first tested positive for bovine tuberculosis in September 2017 and has been in isolation since.
Catrin McGahey QC told the court that although Defra argued in previous hearings that there was a "residual risk" to other animals, the agency has also agreed Ms Macdonald's bio-security arrangements are "impeccable".
She said it had come to light following the publicity resulting from Ms Macdonald's case that other animals who have been subjected to the same testing regime as Geronimo have later showed no signs of the disease after being euthanised.
The barrister said the publicity had led the Daily Mail to find the owners of nine other camelids who were tested under the same regime, whose animals showed no signs of the disease after slaughter.
However, following an adjournment to allow Ms Macdonald's lawyers to decide what evidence they wish Defra to produce and a time estimate of how long that may take, the judge refused to grant injunctions to spare Geronimo pending a further hearing and for disclosure.
Mrs Justice Stacey said that on the evidence before her, Ms Macdonald had not succeeded in showing there was any prospect of her reopening the litigation.
A Defra spokesman said on Wednesday: "There are no plans to execute the warrant today. We are sympathetic to Ms Macdonald's situation, just as we are with everyone with animals affected by this terrible disease.
"It is for this reason that the testing results and options for Geronimo have been very carefully considered by Defra, the Animal and Plant Health Agency and its veterinary experts, as well as passing several stages of thorough legal scrutiny.”