My sister and I can almost always tell, but it also depends somewhat on the texture and/or pattern of hair that has been relaxed.
For instance, no, I did not know that a self described "Italian American" college girl's hair had been lightly relaxed as opposed to straightened because the end result looked just like European-straight hair with a mild wave. There was no way to know that it had undergone a chemical process as opposed to the blow-drying out and flat-ironing that I learned (once living in a dorm with them) many European-ancestry women do daily.
IMO, the tighter the natural curl or kink OR, alternately, the thicker or coarser the texture, I think it becomes much easier to distinguish chemical processing.
However, I do not think *most* people--and certainly not most men, as per that post--have any way to distinguish that. My sis and I grew up analyzing hair like hawks because we're from a very mixed ethnic background where people are always gossiping about "whose hair is naturally that 'good" (quite common for us) and whose hair had to be processed such (also common), and who just had an excellent stylist who knows their way around a hot comb.