What made Dunkirk possible?

Could another navy have carried out the Dunkirk evacuation?

Potentially, but only states with enough naval and merchant shipping, plus the willingness and geographic opportunity to use them, could have attempted it.

Evacuating Dunkirk required available naval power, merchant-marine capacity, motivation, and opportunity. The United States, Japan, and to some extent Norway had relevant maritime capacity, but capability alone would not create the operation. Britain was particularly well placed because its navy and civilian shipping could work together across a short sea crossing. An evacuation is a logistical and political operation as much as a tactical one.