Is it permitted to eat pickled vegetables prepared by a non-Jew?
The general prohibition known as Bishul Akum forbids partaking of foods prepared by gentiles. Even if the food is unquestionably kosher, if a gentile turned on the fire and cooked the food it is forbidden for consumption. Now a fundamental principle in Halacha establishes that "Kavush Harei Hu Ke'mevushal" – pickling has the same status as cooking. Halacha considers the process of soaking a food in water or vinegar for a twenty-four period equivalent to cooking. Perhaps, then, we should forbid pickled vegetables prepared by gentiles on the grounds of Bishul Akum; since pickling is equivalent to cooking, if a gentile pickles a vegetable it should become forbidden like any food cooked by a gentile.
In truth, however, this is not the case. Although in other areas of Halacha we do, indeed, equate pickling with cooking, this equation does not apply to the prohibition of Bishul Akum. This prohibition forbids only foods that were actually cooked by a gentile, and does not include foods that underwent a process of pickling through the hands of a gentile.
One might, however, argue that we should still forbid vegetables pickled by a gentile because of an entirely different issue, namely, the concern that the utensil in which he pickles the vegetables had been previously used in the preparation of non-kosher food. If non-kosher food was cooked in this utensil, the food's taste will be emitted during the pickling process and absorbed by the pickled vegetables. Perhaps, then, we should forbid the consumption of foods pickled by a gentile because it may likely contain the taste of forbidden food.
Chacham Ovadia Yoseph rules that this need not concern us, as we may presume that the utensil used for pickling had not been used with non-kosher food within the previous twenty-four hours. Absorbed taste in a utensil after twenty-four hours becomes Pagum, or foul-tasting, and, according to Halacha, can no longer render a food non-kosher. Therefore, even had the utensil been used for non-kosher food at some point in the past, whatever taste absorbed in the utensil has most likely become Pagum, and therefore has no Halakhic effect on the vegetables pickled in this utensil.
In conclusion, then, one may eat pickles and other pickled vegetables prepared by gentiles.