Yeah, Catholic University of America does provide scholarships based on merit. Here are some of them:

Archdiocesan Scholarship
Minimum Criteria: Students must be at the top of their high school class with at least a 3.8 unweighted GPA in a highly demanding curriculum
Requirement for maintenance: 3.20 cumulative grade point average
Separate application needed: No

The Catholic University Scholarship
Total Worth: $10,000 to $25,000
Eligibility: $10,000 to $25,000
Minimum Criteria: Candidates must have an exceptional academic profile as determined by the quality of their high school curriculum, GPA, and class rank
Separate application needed: No

Catholic University of America provides aids at an average of 31104 per student meeting 82% of students need. 100% of students with need receive financial aid at an average of 27861 need-based grant. They also provide other scholarships including Parish Scholarship and Blessed Saint John Paul II Tuition Scholarship.