“I did not run for Congress to be silent,” the Democrat representative for Minnesota tweeted, before vowing “to fight and to defend our democracy”.
Ms Omar, who was one of the first two Muslim women elected to Congress, said: “No one person – no matter how corrupt, inept, or vicious – can threaten my unwavering love for America.
“I stand undeterred to continue fighting for equal opportunity in our pursuit of happiness for all Americans.”
The congresswoman has faced increasing attacks since the US president shared footage showing Ms Omar referring to 9/11 as “some people did something”, adding the caption “We will never forget!”.
Republicans have accused her of downplaying the terror attacks, and Mr Trump and his son Donald Trump Jr have both either retweeted or liked comments describing Ms Omar as a “sick monster” and “not American”.
In her response, Ms Omar wrote: “This country was founded on the ideas of justice, of liberty, of the pursuit of happiness. But these core beliefs are under threat. Each and every day.
“We are under threat by an administration that would rather cage children than pass comprehensive immigration reform.
“An administration that would rather give billionaires tax breaks than provide a little cushion for working people. An administration that would rather attack fellow Americans who are transgender and wear our country’s uniform than fight for equality and opportunity for all.
“I did not run for Congress to be silent. I did not run for Congress to sit on the sidelines. I ran because I believed it was time to restore moral clarity and courage to Congress. To fight and to defend our democracy.
“Thank you for standing with me – against an administration that ran on banning Muslims from this country – to fight for the America we all deserve.”
“The president is inciting violence against a sitting congresswoman and an entire group of Americans based on their religion,” Ms Warren said. “It’s disgusting. It’s shameful. And any elected leader who refuses to condemn it shares responsibility for it.”
Texas Democrat Beto O’Rourke said: “The president’s actions are an incitement to violence against representative Omar and muslim Americans across the country. There is a cost and consequence to this rhetoric. Members of both parties must stand together and condemn the president’s dangerous actions.”
Mr Trump's actions come just days after a man was arrested for phoning up Ms Omar’s office and threatening to “put a bullet in her skull”.
Ms Omar has previously been accused of antisemitism over comments she made about the pro-Israel lobbying group American Israel Public Affairs Committee.