You may need to be more specific. I think we can say with reasonable certainty that if someone doesn’t die of anything other than X, then X must be what kills them.
In the extreme case, if we allow “anything other than X” to include either atomic decay, heat death of the universe, or the “big crunch” as appropriate, then your statement is a certainty: If “the end” doesn’t kill them, then something else must have, and we’ve disallowed everything except heart disease and cancer, so it must be one of them.
Alternatively, you might be asking: “Given that we could eliminate all causes of death other than heart disease and cancer, would immortality be possible?” I still think it would be a hard question to answer: certainly there’s some nonzero chance of never acquiring HD or C in a “standard lifetime,” (certainly people die at very old ages of other things: accident, stroke, Alzheimers, pneumonia…). But it seems that over “infinite” time, yes, you’d be guaranteed in a practical sense to get whatever diseases were left.
Another interesting possible take on your question: “If we could eliminate all causes of death other than HD & C, what would average life expectancy be?” That might have a factual answer, which I don’t know but would love to hear.