I'm assuming you want to know if the two words have the same root because of the current cultural and political differences between The West (meaning modern-day Europe and the U.S.) and the Arab World. If that is the case (please correct me), the brief answer is no.
I'm going to give a brief background to the readers to explain why your question is a very good one, and then answer your question in more detail.
Please forgive my improper transliteration and the very long answer!
The word غَرْب (gharb) indeed means "west" or, more accurately, "in the western direction". It's such an important word in Arabic and Islamic culture that its form المَغْرِب (Al Maghreb) can mean "the prayer at sunset" and "the direction the sun sets".
The word غَرَبَ (gharaba) means "became estranged" or "departed". If you're feeling slightly poetic, you can say غَرُبَ (gharuba) which can mean "departed towards the west". There is also غَريب (ghareeb), which means "stranger" or "not from our people".
All these words (along with many, many more) have the root غ ر ب (gh-r-b). Of course, they can mean slightly different things in different contexts, as Arabic is heavily dependent on context.
To answer your question (that I assumed), I'm going to use the time period of the early Islamic expansion as my example. Here's why:
It is a very well-documented period of time and it was a little before Europe became important to the Arabs.
Arabic was beginning a very long transition from classical Arabic into Modern Standard Arabic. This was the period where the classical form of the language was being developed in order to study and analyze the Quran and make it accessible to the wider public and future generations. Therefore, the spoken and written Arabic of the time was of critical influence on its own evolution.
"The West" as a proper noun was exclusively used to describe Al Maghreb (the North African territory stretching from west of Egypt to the Atlantic) from this period up until the modern day.
Therefore, I'm going to use this time period to explain why "The West" and "being strange" or "being the other" are not tied together in Arabic, other than that they share the same literal root.
During this period, it was all about the Byzantines, the Sassanids, (who happened to be more North than any other direction) and the expansion into North Africa. "The West" in this case was not an entity, but simply a direction. The only foreign entities to the west that would have been important enough to Arabs at the time was the Kingdom of Aksum بِلاد الحَبَشَة (Bilad Al Habashah) and Coptic Egypt مِصْر القُبْطية (Misr Al Qubtiyyah).
After Constantinople's grip of the levant was weakened, their territory in Africa was wide open and ripe for the picking. Coptic Egypt was the opening to Africa, and after that, westward expansion was natural. Eventually, due to the efforts of Musa Ibn Nusayr, the indigenous peoples assimilated and converted to Islam, cementing North Africa as part of the wider Islamic people well into today. From then on, everything west of modern-day Egypt and Sudan was referred to as Al Maghreb. Everything east was referred to as Al Mashreq (The East), albeit less commonly.
During a time when Arabic, a mostly a spoken and very seldomly written language, was beginning its transformation into a formal language that includes a proper grammar and vocabulary, the west was merely a direction. At the time, its only usage as a pronoun was to reference Al Maghreb (the direction of the sun set) - a part of the Arabs' own territory. To further drive the point, classical Arabic reached its apex and final form by the 13th century - well before the U.S. existed and at a time when Europe was not "The West", either in direction or in title. Therefore, the two words are not linked together in anything beyond having the same root.
I hope this answers your question!