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

St. Francis Xavier Scholarship
Total Worth: full tuition
Eligibility: full tuition
Minimum Criteria: GPA:3.75, SAT Score:1360, ACT Score:31

National Merit Scholarships
Total Worth: $1,000-$2,000
Eligibility: $1,000-$2,000

Miguel Pro Scholarships
Minimum Criteria: GPA:3.0, SAT Score:1070, ACT Score:23

Xavier University provides aids at an average of 24999 per student meeting 67% of students need. 100% of students with need receive financial aid at an average of 22459 need-based grant. They also provide other scholarships including Xavier Community-Engaged Fellowship and Edgecliff McAuley Art Scholarships.