Recep Tayyip Erdogan was speaking a day after PM Ahmet Davutoglu, who had largely negotiated the EU deal, said he was stepping down.
Reports said Mr Davutoglu also opposed Mr Erdogan`s plan to give more power to the presidency.
Mr Erdogan said this was a national need not a personal requirement.
The EU has asked Turkey to revise its anti-terrorism laws as a condition for allowing Turkish citizens visa-free travel within the grouping for short stays.
The EU wants Turkey to narrow its broad definition of terrorism to match tighter EU standards. It is one of five EU criteria Turkey still has to agree to in order to meet the visa-free requirements.
But Mr Erdogan rejected this, insisting Turkey remained under a severe threat of terrorism.
"We will go our way, you go yours," he said in a speech broadcast live on television..
Turkey`s Erdogan takes tough EU line after PM quits
07 May 2016
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Turkey`s president has told the EU it will not change its anti-terror laws in return for visa-free travel, saying: "We`ll go our way, you go yours".
