As an adolescent, I was a member of the San Diego Navy Sailing Club. Once I sailed over from North Island (south of the amphib base) to the 32nd Street docks and got very close to Oriskany in a little 14-foot Capri. This was the mid-1970s. No booms, no interceptions, no verbal warnings. But while I did get very close to the ship, I had enough respect for boundaries not to get too close.
A “suitcase nuke” with all of the handling gear and firing box weighs over a hundred pounds and wouldn’t readily fit inside of a bulkhead compartment of a kayak. If you have a tactical nuclear weapon, you don’t need to get closer than 100 yards.
Stranger
The difference might also be because of the waters they’re in. If the vessels are in a navigable river, then a 1 km exclusion zone is probably going to be completely impossible, since it would completely plug up the river. Given that naval vessels do sometimes use rivers, you therefore couldn’t have a general regulation setting that large an exclusion zone. On the open ocean, though, an exclusion zone that large is easily doable, and therefore done.
Exclusion zones - reminds me of the Imax movie “Mission to Mir”. You see the opening scenes of them rolling a Soyuz out to the launch pad, and a bunch of casual bystanders wandering arund the rail car and alongside the rocket.
Contrast this scene with my visit to Cape Canaveral, where they keep you inside the bus except for one stop, still a few hundred yards from the shuttle launch pads the tour guide warns us that if we step across the line a bunch of armed men in black helicopters will swoop in and arrest us.
I suppose when it comes to security, the USA over reacts and errs on the side of caution. Probably not a good idea to appear to be taunting them.
If you can get your hands on a suitcase nuke, you can probably figure out how to customize your kayak to hold it inconspicuously. 
But you’re right about the range, of course.
If you had a suitcase nuke, why not put it into a water and pressure resistant container and swim underwater to the carrier using some form of scuba gear or rebreather system?
The thing I find amusing about those regulations is the concept of “minimum speed”. How does that apply to a kayak : “I swear, officer, I was paddling as lightly as I could!”
Don’t forget the relevant article on Anti-frogman techniques.
I was on the USS Truxtun (CGN-35) years ago as a midshipman, and once we were anchored in the bay in San Diego, running battle station drills. I’m sure that these drills were quite involved for some people, but for the gunner’s mates I was stuck with, it pretty much involved manning the .50 caliber machine guns wearing helmets and flak jackets, all afternoon long.
To pass the time, they would aim and track the machine guns at passing sailboats and motorboats, making bets on how long it took the occupants to take notice and skedaddle. 
Usually multiple machine gunners took part in this.
Prior to 9/11 I also sailed real close to the carrier (Nimitz, I believe at the time). Closer than 100 yards, I believe. So close, we were in the wind shadow. (Carriers are BIG!) Couldn’t tell if they were pointing the weapons at us. When we went post-9/11, they had the floating barriers around the carrier.
In Oct 2001, we rode jet skis up the Colorado to the Hoover Dam. They had patrol boats on the river that shooed us away. Couldn’t even see the dam around the corner. Must have been a boring posting. About a year later, we went back. Got right up to the floating buoys in front of the dam. Pretty cool.
Minor hijack, please (but relevant, I think).
In 1988 while vacationing in Acapulco, I was thrilled to see that the carrier USS Independence was ‘docked’ there. Never expecting to get an affirmative response, I asked one of its sailors, a super-friendly kid coming back from shore leave, if he could get my wife and I on board. “You bet”, he said and he was good to his word*. Maybe I’m easily ‘wowed’, but it still ranks as one of the highlights of my (admittedly boring) life.
In any case, nowadays, i.e. post 9/11, can you still get taken aboard US Navy ships ‘just like that’?
*even after I confessed that I wasn’t a US citizen (I’m Canadian) and had asked, “aren’t you worried that I might be a spy?”
I can’t imagine getting hosed down would be a deterrent to somebody in a kayak…
Maybe if you approach a Canadian naval vessel?
It could easily capsize you.
Many countries Naval warships when in foreign ports have days when parts of the ship are open to the public. The USN? I don’t know.
In my naivety, I wouldn’t have thought the U.S. Navy could legally shoot a kayaker in Spanish waters.
What the USN can do in a foreign port depends (obviously) on the laws of the country concerned. There could be “visiting forces” legislation under which the US forces have some right to defend their installations but, absent that, they would have no such right.
When the US navy visits Fremantle (where I live) exclusion zones are maintained by, or at least in co-operation with, the local police. Paddle too close to a US aircraft carrier and you won’t be shelled by the US navy; you’ll be hailed by the W.A. water police. It seems to work fine.