The Football Associations of England, Scotland and Montenegro are the only ones to apply the rule, which is intended to boost crowds and ticket sales at matches.
"Out of respect for the wishes of our partners, we will for the time being no longer show matches during the Saturday afternoon blackout period," Eleven Sports said in a statement.
But Eleven Sports, which launched two new sports channels in August for viewers in Britain and Ireland, said the rule which dates back to the 1950s was unfit for the digital era.
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"The blackout is one of the biggest generators of piracy in the UK. These games are easily accessed on illegal sites and it is naive to think fans do not watch them because they are not shown on legitimate platforms, except betting sites," the company added.
"It is irresponsible to leave the market in the hands of criminals... fans in the UK should have the freedom and the choice to watch these games legally through the official rights holder, as they do all over the world."
Eleven Sports said they were considering all legal and regulatory options, including the referral of the case to the appropriate authorities.
(Reporting by Nivedita Balu in Bengaluru; editing by Martyn Herman)
