It would depend on a great many factors. For one, would you be wearing the dress of a modern day individual, such as what you are wearing now? Would you be transported as you are, essentially, or to blend in?

Furthermore, would you suddenly develop the ability to speak the local tongue, or any tongue of the ancient world?

Roman Slavery was not at all based on race, as slaves were drawn from all over the rest of the Mediterranean World, including Gaul (France), Germania (Germany), Hispania (Spain), Britannia (Britain), Greece.^1,2

As far as based on your appearance besides race? That's a difficult matter. Seneca states that, "a proposal was once made to distinguish free men by their dress...it then became apparent how great would be the impending danger if our slaves should begin to count our number".^3 Bradley seems to draw from that that slaves were not easily distinguished by physical appearance, a conclusion that I agree with.^4 Here is an excerpt from Bradley's article:

"The incident Seneca describes (which there is no cause to doubt as a matter of fact) suggest that in normal circumstances Roman slaves were not easily distinguished by physical appearance alone, and other items of evidence could be invoked to substantiate that inference: Appian, for example, in a moralistic digression on contemporary Roman decadence lamented that "slaves are dressed in the same fashion as their masters" and that "except in the case of senatorial rank the same costume is common to slaves and to free citizens." Moreover, if Roman slaves are compared with their counterparts in the slave societies of the New World, it is certainly true that the most important means of differentiating between slave and free in the Americas, skin colour, was almost completely lacking at Rome: the huge contrast between black save and white master simply did not exist and so a contrast between inferior black and superior white could not be drawn."^4

So, to conclude, it would depend. Race would not be the primary factor, but if you were dressed in strange clothes, could not communicate with anyone, and generally ran amok, you might draw suspicion from the locals.

If, however, you could speak the language or you were in the dress, you might already be assumed to be a slave or a citizen. It is important to note that most slaves were free to walk about, especially Greek slaves, who were often tutors. They were not constantly chained up or anything of the sort.

So, in the end, I would say no. You would not have been enslaved right away.

Footnotes:

1. Appian, Civil Wars 2.120; Seneca, On Mercy, 1.24.1.

2. Bruce W. Frier and Thomas A.J McGinn (2004), A Casebook on Roman Family Law, p.15.

3. Stefan Goodwin (2009), Africa in Europe: Antiquity into the Age of Global Expansion, p.41.

4. Keith R. Bradley (1988), "Roman Slavery and Roman Law" Historical Reflections vol 15, no 3. p 477. (The whole article is worth a read).

Edit: I also wanted to source a post I made about a year ago on the topic of slavery here. You might find a great deal more information there regarding the nature of slaves, as well as some pointers for reading up on it!

Hope that helps!