As with most questions about history, there is no single yes-or-no response possible to this one. But the in-between answer I will give leans more toward "no" than "yes."
Both the lengths of time throughout history during which Europeans have controlled the Middle East and the extent of their control need to be taken into account. In the fourth century BCE, Alexander the Great conquered in a short time an enormous stretch of territory from the Bosporus to the Indus river in western India. His premature death meant the huge conquered territory was divided among his generals. Some areas reverted to Asiatic control relatively quickly; others did not. Parthia (present-day Iran) became independent of the Seleucid Empire (ruled by Alexander's successors) in 247 BCE. If we consider Palestine and Syria, however, these remained, with notable exceptions such as the Maccabean period, under Greco-Roman-Byzantine control until 636 CE, when they were taken over by the Arabs.
Apart from the brief periods of control by the Franks (Western Europeans) during the Crusades, it was not until Napoleon's period that Europeans again were to establish a foothold in the Middle East, first in Egypt and then in Palestine. The French were quickly expelled, but this began a period of indirect control of the weakening Ottoman Empire, primarily by Britain and then finally, between about 1900 and the end of World War I, by Germany.
With the Allied victory in 1918, the Ottoman Empire was disassembled, and the Arab territories became European protectorates. This meant that they were essentially European colonies, the British having broken their half-promises about independence to the Arab leaders who had fought on the Allied side to expel the Turks. But after a certain period of years the protectorates were to become independent. Palestine, Jordan, Egypt, and Iraq came under British control; Syria and Lebanon, under French control. By the 1930s and 1940s, independence was granted to all these states, but the Europeans continued to exercise a degree of control in the oil-rich region through puppet leaders and economic leverage. In Egypt and Iraq in the 1950s, revolutions occurred in which the Arab monarchies set up by the British were overthrown, and socialist governments—often backed by the Soviet Union—were installed. In the early 1950s in Iran, the socialist government of Mohammed Mosedegh, which overthrow Shah Reza Pahlavi, a Western puppet, was then defeated by a counter-coup instigated by the CIA and British Intelligence, putting the Shah back in power for another 25 years until he was finally expelled by the Iranian Revolution of 1979.
The pattern through history—in antiquity, the Middle Ages, and the twentieth century—has been that Europeans have taken over the Middle East, held it for a period of time (parts of it even up to several hundred years), and then were driven out. The cultural influences of the Europeans, such as the Hellenistic features of the Middle East for several centuries after Alexander's conquests, have eventually been eroded, and the regions' native cultural character largely restored. In the twentieth century the period of direct European control did not last more than 40 years, starting in 1917–1918. By the 1960s, Syria, Lebanon, Israel, Jordan, Iraq, Saudi Arabia, and Egypt were fully independent countries. The developments since the Iraq War of 2003 have resulted in foreign control of much of the area, but this time, it was led by Americans, with only limited support from Britain and other English-speaking countries. In sum, European and US control over the Middle East has had limited success in terms of length of time and any attempt to impose Westernization on their societies. It has been an on-and-off affair not leaving deep roots and generally not lasting long.