He was speaking ahead of the first vote by MPs on the UK Internal Market Bill, which has sparked threats of legal action in Brussels and outrage at home as it overtly seeks to breach global law.
"I simply can not approve or endorse a situation in which we go back on our word, given solemnly", Cox said Monday on Times Radio.
Ireland's Fine Gael MEP Seán Kelly said the level of opposition, with the five prime ministers who preceded Johnson loudly critical, means the law may not pass in time for the UK Government's end of year deadline.
British Prime Minister Boris Johnson's plan to breach the Brexit treaty breaks Britain's tradition of upholding the rule of law and pushes it into "the ranks of despots", an ally of German Chancellor Angela Merkel said on Tuesday.
"Lawmakers should not be lawbreakers" - A Tory former Cabinet minister has said it would be "unacceptable" to breach worldwide law with legislation to override the Brexit divorce deal.
Johnson accuses the European Union of threatening to use the withdrawal treaty agreed in January to put up trade barriers between mainland Britain and Northern Ireland, and even to impose a food blockade, the latest brinkmanship of a four-year saga since Britain voted narrowly to leave the bloc.
Labour's amendment to block the United Kingdom Internal Market Bill from receiving a second reading was defeated earlier by 349 votes to 213, majority 136.
Every living former prime minister - Conservatives John Major, David Cameron and Theresa May and Labour's Tony Blair and Gordon Brown - also warned of the risk to the UK's global reputation.
Johnson acknowledged on Monday that "some people will feel unease over the use of these powers - and I share that sentiment myself".
Cox, previously a strong supporter of Johnson on Brexit, said he wouldn't support the proposal when the House of Commons takes its first vote on the contentious legislation later Monday.
Last week the European Commission gave Johnson until the end of September to withdraw the bill and threatened to take legal action against the United Kingdom, with Commission Vice-President Maroš Šefčovič describing the bill as an "extremely serious violation" of the Withdrawal Agreement which had "seriously damaged trust between the EU and the UK".
Andrew Mitchell said he backed large parts of the Bill, but would not back it unless it was amended, stating: "To ask members of the parliamentary party to walk through the Lobbies quite deliberately voting to breach worldwide law is something which I can not do".
British Prime Minister Boris Johnson secured initial backing for his controversial plan to breach the Brexit treaty agreed with the European Union past year, despite outrage in Brussels and anger at home over such an overt move to break worldwide law.
"Either he was not straight with the country in the first place or he did not understand it", said Miliband. Both sides accepted the compromise to protect the open border, which helps underpin the peace process in Northern Ireland.
One of Johnson's special envoys, MP Rehman Chishti, earlier quit over the bill - and more trouble could be ahead. I'll very happily give way to him.