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

George W. Jenkins Scholarship
Total Worth: full cost of attendance plus stipend for books, transportation, and personal expenses.
Eligibility: full cost of attendance plus stipend for books, transportation, and personal expenses.
Minimum Criteria: GPA : 3.75
Requirement for maintenance: maintain 3.0 GPA
Application deadline: November 1

Stamps Scholarship
Total Worth: full cost of attendance plus an enrichment fund stipend
Eligibility: full cost of attendance plus an enrichment fund stipend
Minimum Criteria: Based on academic performance
Requirement for maintenance: Maintain 3.0 GPA
Application deadline: November 1

President’s Scholarship
Total Worth: $18,000-$28,000
Eligibility: $18,000-$28,000
Minimum Criteria: Based on academic performance

University of Miami provides aids at an average of 46432 per student meeting 96% of students need. 99% of students with need receive financial aid at an average of 32486 need-based grant. They also provide other scholarships including Stamps Family Charitable Foundation Music Scholarship and National Merit Scholarship.