Cruz called for replacing the current tax system with a flat tax.
"It ought to be just a simple one-page postcard and take the agents, bureaucracy out of Washington and limit the power of government," he added.
Lawmakers are probing the IRS's use of higher scrutiny for Tea party groups seeking tax-exempt status.
Cruz said the scandal highlighted the extent to which the White House would use its powers to go after political opponents.
"The Obama administration has demonstrated a willingness to use the machinery of government," said Cruz.
The Texas senator has been one of the most outspoken proponents of overhauling the tax system. In May, after news broke of the IRS scandal, Cruz tweeted that the only way to prevent that conduct from happening again was to get rid of the agency.
Mr. President, if your #1 priority is fixing the problem, let's abolish the #IRS and ensure it NEVER happens again!— Senator Ted Cruz (@SenTedCruz) May 16, 2013
The new chief of the IRS, Danny Werfel, testified before Congress on Monday, vowing to correct the problems within the agency.
"We have a great deal of work ahead of us to review and correct the serious problems that have occurred at the IRS and continue the important work of the agency on behalf of taxpayers," Werfel said.