Wasn’t von Trapp a captain in the Austrian navy? 
Yes, and a pretty good one. Author John Biggins wrote some enjoyable adventure novels about the Austro-Hungarian navy some years ago.
Between 1949 and 1967, when Israel held the western bank of Lake Tiberias, AKA the Sea of Galilee, and Syria held the eastern bank, Israel kept a small force consisting of 3 armed LCM-6 landing craft on the lake, to protect Israeli fishermen and prevent incursions. I don’t think the Syrians had any craft of their own.
The unit was decommissioned after the Six-Day War.
I’ve sailed from Toronto to Niagara-on-the-Lake, and back again, a few times. The only government vessels I’ve seen on Lake Ontario are Coast Guard vessels (Canadian and American Coast Guards). Nice to know that they are both there to help us, if we needed it.
The only time I’ve seen Navy ships on the Great Lakes (US and Canadian navies) is when they’re on a goodwill tour. I got to tour the HMCS Athabaskan and the USS William C. Lawe, in Toronto, when they were both on a goodwill tour back in 1985 or so. I will say, as a Canadian, that I found the American sailors of the Lawe to be quite polite, quite approachable, and willing to answer questions. Ours aboard Athabaskan were too, but I think my friends and I got a bigger kick out of visiting a US Navy vessel than one of our Canadian ones. Even though we’d never been aboard a commissioned Canadian one.
Edit: A little research tells me that I probably toured the Lawe in the late 1970s. She was decommissioned in 1983, and used as a target vessel in 1999.
Ohio, Wisconsin, and New York all maintain naval militias as part of their organized militia / state defense forces.
The Ohio Naval Militia’s major routine mission is patrolling the Lake Erie coast near Camp Perry. Big chunks of the surface danger area for the Camp Perry small arms ranges extend out into Lake Erie. Someone has to shepherd people that are trying to get themselves shot away.
This OP is littorally the best navy question on the Dope.
There is always the Mongolian navy (which consists of one boat on a lake):
In less friendly times, Lake Champlain was twice the cite of key naval engagements between American and British forces during the American Revolution and War of 1812, respectively. In the Battle of Valcour Island, Benedict Arnold essentially built a fleet from scratch to check a British advance down the Hudson. The Battle of Plattsburgh/Battle of Lake Champlain halted a British offense toward the end of the War of 1812 which impacted the British bargaining position in the ongoing negotiations in Ghent.
Site?
The site is my cite. 
Excellent. I love this cyte.
Not exactly a lake, but as a large navigable body of water, the US Navy did operate gunboats on the Mississippi River during the Civil War and various support craft were built on yards up and down the river for use in WWII.
These days, it’s primarily the Coast Guard (if any DoD division is counted at all) that has operations on the river itself with the Navy having some stuff closer to the mouth of the river.
Haven’t looked but I wouldn’t be shocked if there was at least some naval presence on the Danube (possibly by several nations) which feeds into the Black Sea.
The British and German navies fought several battles for control of Lake Tangayika in Africa during World War 1. It’s a fascinating history. The Germans brought their largest ship to the lake by disassembling it into thousands of pieces and then reassembling it on the lakeshore. The British brought their ships in whole, first by rail and then oxcart through the bush. The British motivation was memorably expressed as “It is both the duty and the tradition of the Royal Navy to engage the enemy wherever there is water to float a ship”
Well, how many lakes large enough to both with warships are shared between two or more countries? You have the Great Lakes shared by the UAS and Canada - mostly Superior, Erie and Ontario - but they’re friendly nations and have agreed not to militarize the lakes.
Lake Victoria is shared by three countries, but I don’t think they’ve bothered to militarize it.
The British also launched the monster HMS St. Lawrence, a 112 gun first rate which was the only Royal Navy ship of the line (i.e. third rate or higher) to serve entirely in freshwater (Lake Ontario).
Now as far as say… the Rio Grande between Mexico and the US? It’s patrolled primarily by the US Border Patrol, and the Texas Department of Public Safety, not the US Navy.
I didn’t include rivers in my original post, but sure, feel free to include them. I imagine a lot of this lake ‘n’ river militarization is historical. But, you know, Apocalypse Now was all about navigating a scary river on a mission.
When we were kids I used to joke with my brother about a made-up TV show called “Lake Patrol!”, where the tough lake cops would bust scofflaws for things like waterskiing after sunset.
That and it’s hardly navigable by a patrol ship of any reasonable size for much of its length.
The explorer who named the Rio Grande discovered it during the flood season, when it is “a mile wide and an inch deep”.
Rivers you say. The US Army [YES! ARMY] operates vessels on our nation’s waterways. The US Army Corps of Engineers operates dredges, research, and patrol vessels. On the lookout for slime creatures, invasive fish, … really wandering off topic here … sorry … slinks away.
Well, if I were the Navy, I’d make sure the Navy Corps of Engineers (is there one?) had a thing or two to say about that! Where are our boundaries?