I'm 99% sure Babcia said this word-for-word. An old friend's babcia could only say two things in English - "Is nice" and "EAT, EAT".
I do not know your friend's babcia, but it is highly probable she used "to" instead of "więc" (I see as slightly more formal, but it could be that it is more common in other regions)
It might be a little old fashioned, but 'He eats not, therefore he is weak' is a correct English sentence and should be accepted as far as I know?
It's so incredibly archaic as to be pointless - nobody would have written or spoken like that later than the year 1650 or so (this is often called "Early Modern English").
'dlatego' is like 'that's the reason why...', and 'więc' is simple 'therefore'.
This is another with multiples in saying, hence is another way with so and others. Most of my errors are in typing, hitting 2 keys at a time and other ridiculous goof ups.
To say: "If he does not eat, he will be weak." would this be possible?
"Jeśli on nie je, więc jest słaby."
(out of interest, not suggesting it as an alternative answer)
No, you can only use one of those for the sentence to make sense: either "jeśli" or "więc".