The 27-year-old Frenchman pleaded guilty to two counts under the Animal Welfare Act when he appeared at Thames Magistrates’ Court in east London on Tuesday morning.
Zouma was said to have meted out his violent retribution on the defenceless male Bengal cat after blaming it for damaging a chair at his home.
Two cats have since been signed over to be rehomed.
The court heard the “premeditated” attack was filmed by his younger brother, lower-league footballer Yoan Zouma, 24, who sent it to a woman he was due to go on a date with.
But the woman was so shocked that she called off their meeting, telling him: “I don’t think hitting a cat like that is OK – don’t bother coming today.”
Yoan Zouma admitted one count of aiding, abetting, counselling or procuring his older brother to commit an offence.
The pair were released on bail ahead of their sentencing at the same court next week.
Trevor Burke QC, for his client, said Kurt Zouma had been subject to racist abuse and threats, lost a “substantial” sponsorship deal with Adidas, was fined £250,000 and had no longer been picked for the France national team since the furore over the footage emerged.
Seven people are to be prosecuted by police over the Zouma abuse.
The court was told about disturbing footage of the incident, filmed at Zouma’s home and posted on Snapchat on February 6.
The Premier League star could be seen volleying the Bengal cat across his kitchen, before throwing a pair of shoes at it and slapping its head.
The clip contained laughing, and featured laughing emojis.
Prosecutor Hazel Stevens told the court Kurt Zouma could be heard saying: “I swear I’ll kill it, I swear I’ll kill it.”
He admitted two counts of causing unnecessary suffering to a protected animal on February 6.
Hammers defender Zouma arrived at court in chaotic fashion, accompanied by several security guards who emerged from a vehicle holding umbrellas to form a protective shield around the defendant.
We hope this case will serve as a reminder that all animals deserve to be treated with kindness, compassion and respect, and that we will not tolerate cruelty by anybody
Dermot Murphy, RSPCA
The brothers’ addresses cannot be published because of concerns over the threats to the pair.
In a statement, RSPCA chief inspectorate officer Dermot Murphy said: “It’s never acceptable to treat an animal in this way and we were shocked when the video was first brought to our attention.
“What makes this case even more sad is the way the video was filmed and shared, making light of such cruelty.
“We hope this case will serve as a reminder that all animals deserve to be treated with kindness, compassion and respect, and that we will not tolerate cruelty by anybody.”