Long story short: The RCN provides NO repeat NO constabulary services. The RCN has no police / peace officer power to enforce any laws. The RCN does have the authority to fight armed intrusions into Canadian waters by a nation state actor - that's it. We are military, period.
The separation of functions in the Canadian maritime environment harks all the way back to the original Department of Marine and Fisheries instituted shortly after Confederation to basically control offshore, inshore and inland fisheries and to provide (in along list) for harbours, wharves, harbour masters, aids to navigation, including all lighthouses, buoys and shore based markers, etc. and for the certification of masters and mates and inspection of vessels. There was no navy at that time and the fisheries side of things obtained "cutters", which were really light cruisers, to enforce fishing rights against the Americans, since the RN -in charge of our coastal protection - refused to do it for fear of another war with the Americans.
This fisheries protection side of the house became the incubator of the RCN, but continued it's existence and function for all times thereafter, with the RCN serving purely as military muscle.
In all this, constabulary duties (the enforcement of Canadian laws in Canadian waters) was , and remains to this day, the duty of the police, and in particular, the RCMP. When RCN ships are sent to carry out law enforcement, they carry RCMP officers onboard to carry out the policing work. For quite a long time, the RCMP had a rather large marine detachment of ocean/inshore capable vessels, but in the late 1960's/early 1970's, they mostly disbanded it. The vessels ended up as Naval Reserve units tenders or even commissioned vessels (HMCS FORT STEELE). The RCMP now uses large RHiB's and a few twin-hulled small fibreglass vessels on the West coast.
Shortly after WWII, the various government fleets grew quite large and, as ships became more and more capable of operating in some ice, ice breaking became a much more important mission for keeping the sea lanes (read the St Lawrence river and the coasts of Newfoundland) open. Thus, the fisheries remained on their own, but the other duties, relating to nav aids, harbour and certification, together with the now increased icebreaking fleet, was spun out to the Department of Transportation. Also, a new department was created, required as ocean research became more and more important both to support fisheries regulation and to provide the Navy with oceanographic data for ASW and navigation: The Department of Oceans and Hydrographic research.
Then, the marine services that were within the Department of Transportation became a specific service within the Dept and was renamed Canadian Coast Guard, but remained centred on its nav aids and icebreaking duties + marine certification / inspections. Marine SAR, which they had previously done on a secondary basis, was confirmed into the mix. Finally, the government decided that it had too many different "fleets" and started, first, by amalgamating the Department of Oceans with the Department of Fisheries -spinning Hydrographic services out totally under the Navy for a short while (that merger between scientists and para-military like fisheries personnel went very well: NOT!). Then, they amalgamated the whole lot under the Coast Guard, so that right now, there are basically three organisations only that oversee all aspect of Canada's maritime environment: Ocean Services (The Coast Guard) for all aspects not covered by the other two; the RCN and it's auxiliaries, for all naval defences and military operations of Canada anywhere in the world; and, the RCMP/police (because some of it is done by provincial and municipal police forces within their territories) for all law enforcement duties - with the RCMP operating a few small crafts and bumming ride with the other two services need be (particularly onboard the CCG's mid-shore patrol vessels).
But the Coast Guard did not take over any constabulary duty - nor has such duty been passed to the RCN: It remains with the police. Although, I believe that some limited constabulary powers have now been entrusted to Border Services Canada and that they have some small RHiB's to carry out these functions.
