He faces arrest if he travels to any country party to the International Criminal Court’s jurisdiction, including the UK.
Following the issue the warrant yesterday, Downing Street said PM Sir Keir “respects the independence of the ICC”.
No10 said it will now be up to domestic courts to make a final determination on whether Britain endorses the decision.
A spokesman also refused to say whether Mr Netanyahu would be welcome in Britain and would not clarify whether he would be arrested if he travelled here tomorrow.
But Shadow Justice Secretary Robert Jenrick said: “The UK shouldn’t enforce this farcical warrant from a politicised court. Starmer is a disgrace.”
Shadow Foreign Secretary Priti Patel added: “Labour must condemn and challenge the ICC’s decision.”
And ex-Home Secretary Suella Braverman said: “The ICC has lost credibility. They’ve made a mockery of the law and undermined the international rules-based system.”
The UK is a signatory of the ICC, based in the Netherlands, and Attorney General Richard Hermer has repeatedly vowed to respect its rulings.
The court also issued arrest warrants against former Israeli defence minister Yoav Gallant and presumed-dead Hamas chief Mohammed Deif.
Its judges said there were “reasonable grounds” the men bore criminal responsibility for alleged war crimes and crimes against humanity.
Israel and Hamas reject the allegations. Mr Netanyahu’s office called it anti-Semitic.
US President Joe Biden’s administration also “fundamentally rejected” the ruling.