I presume you’re asking this because of the big debacle on Twitter a few months ago?
Any way you care to look at it, the Roman Empire was multicultural. This is not generally considered a controversial statement. Frankly, it would be simply impossible for the Roman Empire to be anything but multicultural, given the vast territories it encompassed.
Remember, the Roman Empire did not extend simply within the confines of Europe; it spanned from North Africa to Western Europe. This means a number of things.
that there were people living in North African territories, who were of North African descent, who became citizens of the Roman Empire when they were conquered. These people did not suddenly stop being North African just because they now lived under the Roman Empire; their culture did not die. To take the objection most commonly raised nowadays, the skin colour of these people did not change either. They were not replaced by Roman populations, but retained much of their populace under the Roman Empire, and became a mixed society.
Roman culture was inherently syncretic (i.e. formed of many different source cultures) owing to its territorial expanse. As new territories were conquered, the cultures of the conquerors and the conquered merged to form new religious beliefs, cultural norms and ways of life. What we think of as Roman religion was completely syncretic, formed of elements absorbed primarly from Greek religion and Etruscan religion, and other beliefs were later included as the territories expanded. Egyptian gods continued to be worshipped in Egypt, and were also brought to other places in the Roman Empire, including Rome itself, where the cult of Isis was particularly popular. Roman art was influenced by art from their territories, particularly Greece and Egypt. Similarly, Roman beliefs spread to Egypt, and the depictions of their traditional gods began to include – but were not replaced by – Roman imagery. The cultural exchange worked both ways.
migration became more possible and more common. We have evidence of people travelling across the Empire, from North Africa to Britain. Many influential Roman writers, including Apuleius, a popular satirist, were not from Rome; he, for example, was North African. Several emperors were not of singularly Roman descent; Elagabalus was Syrian, to take a well-known figure. Remains which have been identified as being those of migrants have been found across the Empire. It’s not possible to speculate the extent to which this occurred, as we quite simply don’t have the data, but we know that people did travel and settle across the Empire. This is evident not only in skeletal remains but in the transfer of culture, art, and religious iconography.
the Roman Empire did not simply expand into one huge, single culture. Rather, it deliberately incorporated dozens of other cultures, meaning that what we think of as ‘Roman’ culture was incredibly fluid and always open to change from external factors. Also, conquered territories were permitted not only to take on Roman beliefs and cultural practices, but were able to retain their own. Often, their cultures were influenced by Roman ways of life, but not always, and often they were as influential as they were influenced.
I’m at a loss as to why this is now considered controversial, as it is simply unfathomable that the Romans could have conquered territories across two continents and maintained one monoculture across each and every territory. It simply is not how empires work; even the British Empire was influenced by the cultures it conquered.
By any definition of that term, the Roman Empire was multicultural. We're talking about an entity that spanned millions of square miles, had a population of tens of millions and lasted for over 500 years. How could it be anything else?
As it expanded, the Empire conquered lands inhabited by non-Roman peoples - North Africa, Greece, Asia Minor, Gaul, Egypt, Britain and so on. The people who lived in these places did not immediately stop practicing their previous cultural behaviours in favour of 'Roman' ways of life: they continued to speak their languages, worship their gods, wear their clothing and eat their food. The Roman Empire was literally multicultural, in that many different cultures continued to exist within its boundaries.
In certain places, and among certain groups, its true that some elements of 'Roman' culture did become more widespread. In the western Empire, Latin became the most widely spoken language (Greek was never supplanted in the east), and people did start worshipping Roman gods, eating Mediterranean food and so on. This doesn't mean that these people became 'Romanised' in the traditional sense. In most places, hybrid cultures emerged, made up of a creative mixture of cultural forms from Rome, Italy, the local area and the wider empire. People seem to have adopted and adapted cultural forms as they suited them, and as they made sense to their own individual contexts.
There's little evidence that the Roman state itself tried to forcefully impose this culture on the inhabitants of the provinces - there was no movement to create a monoculture in the Roman Empire. Local cultural forms were allowed to continue, as long as they didn't directly challenge the power and authority of the Roman state.