Would the gull-wing Spitfire have fixed landing gear?

Would the proposed gull-wing carrier Spitfire have retained the test aircraft's fixed landing gear?

No: the test aircraft used fixed gear, but the planned service fighter was to have retractable gear and a gull wing better suited to carrier-landing loads.

The fixed undercarriage belonged to the demonstrator used to explore the wing form, not to the intended operational fighter. The service design was planned with retractable landing gear. Its gull wing would also raise the propeller and help the structure cope with the controlled crash of a carrier landing. That distinguishes the proposal from the later Seafire, whose essentially land-based Spitfire structure absorbed stresses for which it had not originally been designed.