To misquote The Royal Tennenbaums, "Everyone knows JFK dies in Dallas. What this question presupposes is...maybe he didn't?"

I'm going to assume that you're asking if he had survived an assassination attempt, would this have been a cause for war with the USSR? This is the kind of counterfactual that is difficult to assess. What we can do is analyse a) JFK's stance on the Cold War and b) the actual reactions to his death.

Let's look at point 'a' first.

Kennedy oversaw increases in defence spending compared to the Eisenhower years (although as a percentage of GDP it remained quite similar). There are lots and lots of reasons for this, but in general this was a period when Intercontinental Ballistic Missiles (ICBMs) and Submarine Launched Ballistic Missiles (SLBMs) are becoming the backbone of US nuclear strike capabilities (with the caveat that there are still hundreds of nuclear capable heavy bombers in the USAF). These (the missiles) are obviously expensive to design, build, maintain, and create supporting infrastructure for. Then, there was also the relentless expansion of the Cold War in the Global South, etc.

Kennedy was undoubtedly a 'Cold Warrior', as his inauguration speech made clear. He wanted to 'win' the Cold War (but was short on detail as to what this might actually involve) and saw the 1960s as a period of particular danger. JFK was also very ideological, and was genuinely committed to the victory of US ideology over Soviet communist ideology. However - and this is important - his initial belligerence was tempered by event such as the Second Berlin Crisis (when the Berlin Wall was constructed, he was wont to comment "A wall is a hell of a lot better than a war." and (crucially) the Cuban Missile Crisis. He and his counterparts in Moscow came to recognise the existential dangers of Cold war competition, the arms race, etc (a good example is his June 1963 American University speech). Hence the installation of the Moscow-Washington 'hotline', the signing of the 1963 Limited Test Ban Treaty, and the roots of detente in the aftermath of Cuba.

Now, point b.

First off, I'd direct you to the recent question that /u/SilvioBurlesPwny posted on Lee Harvey Oswald's motivations for assassinating JFK. My answers there might provide some useful context.

In the aftermath of the assassination Lyndon B. Johnson was deeply concerned that it might lead to nuclear war. Why? Well, LBJ knew about the CIA plots to kill Fidel Castro and believed that the Cubans had got to Kennedy first. If, he reasoned, word got out that the Cubans had killed Kennedy, then the American public would demand an invasion. This would break the non-invasion agreement that helped to conclude the Cuban Missile Crisis, the USSR would undoubtedly intervene to support Havana, and this World War Three would start. This is isn't speculation, this is what Johnson thought at the time.

And according to archival sources, the Kremlin was terrified that the assassination represented the first stage of a right-wing anti-communist coup in Washington that would lead to a pre-emptive strike on the USSR. So rather than either side seeing the situation as an opportunity for war, there was horror at the thought that one killing might spark global nuclear armageddon.

Given the Johnson administration's - ultimately failed - effort to ensure that the American public were 100% behind the idea that Lee Harvey Oswald - and Oswald alone - was responsible for the killing, it's reasonable to assume that if JFK had been shot and survived, a similar situation might have eventuated. Moreover, Kennedy was quite an ill man already (something largely kept from the public at the time). Serious injuries from gunshot wounds might have necessitated a transfer of power to LBJ anyway, and we're back at the actual situation in late 1963.

TL:DR? We simply don't know what might have happened, all we can examine is what did happen. Washington and Moscow both feared nuclear war in the assassination's aftermath, but events of previous years had really brought home the manifest dangers of the Cold War superpower standoff.

Hope this helps

Malcolm