WASHINGTON—Attorney General William Barr said he would deliver the special counsel’s still-secret report to Congress within a week and vowed to explain his reasons for blacking out parts of the roughly 400-page document, a pledge that did little to mollify demands from Democrats for full access to the report.

“I am relying on my own discretion to make as much of it public as I can,” Mr. Barr told lawmakers on Tuesday, adding that he will color-code the redactions and provide notes explaining the reasons for each.

His...