Long story short, no. But since my focus is on Israeli history, and since debunking JFK theories has happened a lot on this subreddit, let me focus on the nuclear aspect.

It is quite accurate to say that JFK sent a letter in May of 1963. There had been some back-and-forth before that point, but the gist of it was that things started coming to a head around April. In a meeting with Israel's Foreign Minister, Shimon Peres, President Kennedy asked Israel to clarify what its nuclear program was doing, and said "What can you tell me about this". Peres said a phrase that later became infamous as a statement of ambiguous policy, which he claimed was an improvisation: "We will not introduce nuclear weapons to the region, and certainly we will not be the first".

Kennedy took this to mean that Israel would not seek or have nuclear weapons. Peres took this to mean that Israel would not announce or otherwise show off nuclear weapons, and would maintain ambiguity about its program. The disconnect was fairly clear. However, because Kennedy believed that Israel had committed to no nuclear weapons development, he asked Israel to allow visits to its nuclear facilities by inspectors. A request was made to have semi-annual visits to Dimona, the Israeli facility, and to have the first visit take place in May. Israel did not respond, and two weeks later, a follow-up was sent.

On April 26, Israel finally responded, but largely changed the subject. The letter from Ben-Gurion to Kennedy details the threats Israel faces, discusses the "Treaty of Federation" between Egypt, Jordan, and Syria (which would partially contribute, by the way, to dynamics leading into the 1967 War which had thoroughly spooked Israel and led it to launch its preemptive strike later on), and Israel asks for a declaration from both the US and USSR that would guarantee Israel's territorial integrity. In some sense, this was clearly an attempt to divert the subject, but in another, it may have been a signal to Kennedy that barring serious assurances, Israel would not acquiesce to even inspections of its nuclear program, and almost a tacit admission that it viewed nuclear weapons as the only failsafe against the much larger and more populous Arab world surrounding it.

He basically doesn't answer at all on the nuclear issue directly, however.

Kennedy responds about a week later, saying he understands the threats to Israel. He discusses his belief that a declaration with the USSR is unlikely if not impossible, and generally seems to say "We hear you" on the other concerns. He tacitly brings the discussion back to nuclear issues, by referencing that he believes that the development of "offensive systems" of certain types could destabilize the region and be worse for the cause of peace. However, he doesn't directly re-raise the question of inspections, at least not in those words. President Kennedy appeared to be playing the diplomatic game slowly, here. He didn't want to directly press for the visits himself, yet. So instead the US Ambassador raised the issue again, about another week after Kennedy's letter. Ben-Gurion's response appears to have been that without greater security assurances, Israel would not allow inspections, and otherwise would only bargain for the inspections. Kennedy, it seems, was unhappy with linking the two, arguing that Israel's nuclear systems should be inspected without any need for assurances of Israel's security by the US.

Around May 12, Ben-Gurion decided to continue with the same tack in a letter to Kennedy as the previous one. Prime Minister Ben-Gurion, taking a conciliatory tone, expresses again appreciation for the US's efforts to help Israel in the past, as well as its assurance that it is a friend. Ben-Gurion once again reiterates the threats Israel faces, those that Kennedy says he is less concerned that Israel will fall to, and he points to the rallying abilities of Nasser in Egypt. In this, again, Ben-Gurion had some foresight; it was precisely the ability to cajole and rally allied Arab states even when reluctant that helped drive the 1967 War from a point of merely high tensions into outright war; Israel's preemptive strike was largely decided when Nasser managed to cajole Jordan into a mutual defense treaty that would place Jordanian troops (Syrian troops already were subject to a similar provision) under Egyptian command in case of war with Israel. That, and the feeling of a tightening noose around Israel, helped contribute greatly to Israeli thinking in launching a preemptive strike, and Ben-Gurion was calling it out 4 years prior as a threat that Israel took seriously. He also points to the lack of an arms agreement between the US and USSR (a tacit point that the two were not limiting their own nuclear race) and then pivots it to arms given by the USSR to the Arab world. What followed that was Ben-Gurion asking for things he likely knew Kennedy would not give: equivalent armaments to what the USSR was giving the Arab states, a bilateral defense treaty with the US and Israel, and things of that sort. The National Security Archive posits, and I think I'd agree, that this was because Ben-Gurion was signaling precisely how important the nuclear issue was to Israel, and how gravely it viewed its situation and need for some sort of failsafe to always fall back on.

CONTINUED IN A REPLY TO MY OWN COMMENT.