Is Modern Hebrew an Indo-European Language? First, I'm looking at this question as someone who got his Master's in Arabic Linguistics, and read the Pentateuch in Classical Hebrew, and read some Syriac and Talmud. So, very strong Classical Arabic, decent Biblical Hebrew, and working knowledge of Aramaic and Arabic dialects.
In terms of phonology, yes. Modern Hebrew is a cross between European and Sephardic. The Ashkenazic pronunciation (which seems to be the upper register) has the uvular 'r', does not distinguish between pharyngeal and velar voiceless fricatives, and has no voiced pharyngeal fricative (the 'ayin). But the Sephardic influence comes through in pronouncing syllable-final stops as stops rather than fricatives (Ashkenazi "shabos" vs. Sephardic "shabat"), and through the stress on the final syllable. Although I have no idea about Mizrahi pronunciation.
But what about syntax and grammar?
My pet theory - never propagated - was that the Semitic Sprachbund went through three phases, and that recognizable characteristics propagated across language barriers.
Representatives of the first phase would be Akkadian and Babylonian, which is characterized by a stative ("sharaa-ku","I am the king"), a prefix-verb form "iparras", a nominal the case system, and no direct articles.
The second phase is the one we are familiar with in Classical Arabic, Classical Hebrew, and perhaps older versions of Aramaic. Direct articles appear from the pronoun "hallaa", in Arabic "al-?aalim", Hebrew "ha-?olam", and Aramaic "?alm-aa". The stative suffices became possessive suffices for nouns. A suffix-prefix distinction for imperfect and perfect verbal aspect - Arabic "katab-tu", "I wrote", "ya-ktub-u" "he writes". Genitive forms are expressed using the iDaafah - a noun with no direct article followed by a noun with an optional direct article - Arabic "sayf-u d-diin", "sword of the faith", Hebrew "bnei yisra'el" "the children of Israel", or Aramaic "bar naashaa", "the son of man". This phase tends to have an VSO word order, consistent verbal templating, and some remnants of the nominal case system.
Then, the third phase seems to have happened sometime between 500 BCE-1000 ACE. What is interesting is that these changes were seen in Mishnaic Hebrew, Aramaic, the sort of Aramaic-Hebrew mix of the Gemara, and the Arabic dialects. Direct articles stoped being productive in Aramaic, but survived in Arabic dialects and Hebrew. The word order tended towards SVO. The iDaafah was less productive and replaced with an analytic genitive - Hebrew "shel", Aramaic "d-", and Egyptian "bita?". The verbal system relied more heavily on a participial present ("ana raayiH", "ani holekh"), making the perfect serve for past tense and the imperfect serve for other functions (Egyptian "ana bi-fakkir" "I am thinking", "ana Ha-fakkir" "I will think"). Finally, subordinate verbal clauses tended to be nominalized when possible ("ani rotzeh midaber"). Interestingly, in Syriac the present participle became its own verbal category, taking on suffixes.
From what I can tell (and my knowledge of Modern Hebrew doesn't go much beyond watching Netflix with subtitles), Modern Hebrew has an SVO word order, an analytic genitive, a participial present, use of prefix-verb forms for future tense, and suffix-verb forms for past tense. In this sense, it looks a lot like Mishnaic Hebrew, and thus falls into the third phase. If a traditional Jewish upbringing emphasized mastery of the Talmud, and if there is a lot more Talmud to study than there is Tanakh, then the first Modern Hebrew speakers would probably fall back on Mishnaic Hebrew as a reference. This would be reinforced by the fact that Jewish communities often used this form of Hebrew in written correspondence. The only explicitly European feature of Modern Hebrew grammar seems to be the apparent lack of tense/aspect ambiguity in verbs that you see in other Semitic languages - the prefix verbal form means the future, the suffix verbal form means the past, the participial form means the present. Compare this with Deuteronomy's "va-baHar-ta et-ha-Hayyim", "and [you chose/will choose/should choose/choose it or you'll be sorry] life".
However, if we really wanted to see if Modern Hebrew is influenced by European languages, we could do a small experiment.
First, we would see if it has Germanic-style noun compounding. I assume we can say "mayed shel ha-oTo" ("the carborator of the car"), and presumably we could say the more Classical "mayed ha-oTo", but can we say "oTo mayed"? If this is the case, then this would be a direct effect of German and/or Yiddish on Modern Hebrew.
Secondly, Yiddish is known for topicalizing subordinate clauses. Some think this is from Slavic influence, although I have no idea. "du kumst azoy veyt fir dos?" ("You come so far for this") can be turned into "fir dos kumst du azoy veyt?" ("For this you come so far?"). Does this happen in Modern Hebrew? "ata holekh la-ir" -> "you're going to the city", "la-ir ata holekh" -> "to the city you're going"
Apologies offered for incorrect renderings and corrections appreciated.