Without Suez, Gibraltar seals the Mediterranean

Would the Royal Navy care as much about the Mediterranean if there were no Suez Canal?

No; Britain could rely on Gibraltar to close the western entrance when necessary instead of treating the sea as the route to its eastern empire.

The Suez Canal makes the Mediterranean a route to imperial and global commitments rather than a closed regional sea. Remove that connection and Britain would invest much less attention in the theatre. Gibraltar would still let the Royal Navy seal the western entrance when another power became threatening. One geographic chokepoint can transform an entire fleet's priorities.