No, there is no evidence that Modern English 'pig' is derived or even cognate to Classical Latin 'pinguedo'.
Middle English pigge "a young pig" (mid-13c., late 12c. as a surname), probably from Old English *picg, found in compounds, but, like dog, its further etymology unknown. The older general word for adults was swine, if female, sow, if male, boar. Apparently related to Low German bigge, Dutch big ("but the phonology is difficult" -- OED).
Semantically the 13 c 'pigge' drifted from a small swine to all sizes, and modern pronunciation 'pig' has the connotation of fat now since most modern adult pigs are bred to be fat.
In Latin 'pinguedo' is derived from 'pinguis' which means 'dull, stupid, fat'. Semantically, yes, the Latin original. word has drifted into 'fat'.
However, there is no historical evidence of any sound change, if 'pinguedo' were borrowed in any time of history of English (even back to the Roman occupation of Britain), that the sound changes at any of those time would have naturally led to changing the two syllable 'pinguo' to 'pig'.
Here are some details about borrowings and sound changes (I'll be abusing notation in the use of spelling for sound. as it makes things easier to write). What we're looking for is the word 'pinguedo' or other derivations of pinguo' in Classical Latin or any derived variety (dialects of Italian, Spanish, French) from before the time 'pig' first appeared in English. Mid 13th c really says Old French or classical Latin would be an obvious source.
Since word initial 'p' and stressed (or single syllable word) 'i' are fairly stable in English from Old to Modern, they don't rule out a borrowing from Latin or Romance.
What makes an unlikely borrowing is the '-ngu-' /ŋg/ or /ŋgw/ in Latin/Romance. If borrowed, it is unlikely that English would drop the /ŋ/. Every word in OE with word final /ŋ/ (really /ŋg/) maintains it, instead of dropping it as what should happen to get pingue -> pig. This is the primary reason not to believe a relation between the Latin word and the English one.
Proving a negative is much harder than proving a positive (for a positive just show the example, for a negative you have to appeal to a lot of rules that don't seem to work and in linguistics there can always be a one-off exception). And one of this says that the Latin source is impossible, but it since there's no direct positive evidence of a link (written used words), we have to work with likelihood and the strongest data points to 'no connection'.