You refer to an obituary. Do you have a copy of it which you could scan and post for us to take a look? Are you sure that Juli was the last name (and not the first name)?
He likely acquired the name Andrew only in the USA. If his first name was spelled Andro in the old country, this might point to Croatia, Slovakia or Slovenia. In Hungary, the name may have been spelled as András.
A main Hungarian form related to Juli might be Gyula. Even when assuming that the name was András Juli (with Juli being the surname), the transcribers at Ellis Island might have made the mistake to place Juli into the first name field.
Have you finecombed the Ellis Island arrivals at web site www.ellisislandrecords.org
You should also look at the website https://familysearch.org
. Select Central Europe, Hungary, then the catholic birth records. Enter Juli into the last name field and mark it as precise match. There are quite a number of records although no suitable András/Andreas pops up. But it shows that the possibility exists.
The first name of the gggrandmother might have been Janka (Joan) or Térka (a nickname for Therese).