Trump called out Schiff in a tweet, saying the Congressman was "sleazy" and "totally biased." He also said Schiff spends his time on television "pushing the Dem loss excuse."
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During an interview with Eyewitness News, Schiff said Trump spends too much time watching television and the insults on Twitter are "beneath the dignity of the office."
"It disturbs me to see just how low this office has become under his stewardship, where you have the president of the United States using Twitter to hurl insults at people," he said.
Schiff added that he felt like a character in the movie "Ghostbusters" who had been slimed after reading the tweet.
The Congressman also discussed the three-hour closed session of questioning for Jared Kushner. He said it seemed Trump's son-in-law answered most of the committee members' questions.
"I do feel like he answered our questions. I think it was very productive. All the members that wanted to participate had very good questions for Mr. Kushner," he said.
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Kushner, along with Trump's former campaign chairman Paul Manafort, met with the committee behind closed doors to discuss meetings with Russian officials, which sparked the Russia probe.
Both men had cooperated with the committees which, along with Special Counsel Robert Mueller, are investigating Russia's interference in the 2016 election and possible collusion with Trump associates.
The two men have faced particular scrutiny about attending a Trump Tower meeting with a Russian lawyer because it was described in emails to Donald Trump Jr. as being part of a Russian government effort to aid Trump's presidential campaign.
There have also been reports that Trump wants to remove Mueller as special counsel. Schiff said he found it "appalling" and that he and Congress would not let Trump interfere in the investigation.
The Associated Press contributed to this report.