Based on wikipedia, modern Greek developed directly from ancient Greek. However, modern greek has some words and phrases that are very similar to romance languages. During the time of the Roman Empire, when both Latin and Greek were spoken, did the languages "merge" in any way?
I could be horribly misinformed about any of the assumptions I made in the above paragraph, so feel free to correct me.
First, if you don't get a good answer you may want to try /r/linguistics. While we do have a few linguistics experts here, they've obviously got a much broader base of experts in various fields of linguistics.
However, a point on your OP. Words and phrases being similar doesn't mean languages "merged". Languages borrow words from each other all the time. Languages merging implies that the languages fused their grammar. While this does occur (creoles are an example, such as Haitian Creole), it's a different phenomenon than adding loanwords, even in large numbers. For instance, English was massively influenced by Norman French--a significant proportion of English vocabulary is from there. And Yiddish was massively influenced by Semitic and Slavic languages--depending on dialect, huge swaths of vocabulary aren't Germanic. Yet neither one "merged" with those other languages--they were influenced, but their structures remained distinct. While I can't answer the specific question of Latin and Greek, it's a point that's important to make to understand how languages influence each other.
Additionally, Greek and Latin are related languages. Both are Indo-European languages, so one would expect great similarities in vocabularies and grammars even if the languages had never had contact.