The tech boss insisted the social media app was within its rights to limit the US president's posts on the Discover page.
He told CNBC's Power Lunch: "We've always said Discover is a closed platform, and we choose the types of content we want to promote on our platform.
"We're well within our First Amendment rights to decide what shows up on there."
Last week, the company decided to stop promoting Trump's content through the Discover feature - although users can still follow the president and view his posts.
Trump's campaign chair Brad Parscale argued in a statement: "Snapchat hates that so many of their users watch the President's content and so they are actively engaging in voter suppression."
However, Spiegel has fired back and defended the company's stance.
He explained: "The First Amendment is very specific. It's actually designed to protect individuals and private businesses from the government... We want to use our rights to stand up for the things we believe in."
In a previous statement, a spokesman for Snap said: "We are not currently promoting the President's content on Snapchat's Discover platform.
"We will not amplify voices who incite racial violence and injustice by giving them free promotion on Discover.
"Racial violence and injustice have no place in our society and we stand together with all who seek peace, love, equality, and justice in America."