Did Kennedy want the US Navy to destroy Soviet Subs?

No

Explanation:

The initial impression of Kennedy was to end the standoff through diplomatic means with military invasion as the last resort even though our forces were planned already for such eventuality.

The navy was to institute a blockade to the Soviet Subs so as to avoid further infiltration by the Soviet forces to the already installed Nuclear facility in Cuba while still following diplomatic means from the Soviet government to remove the installation.