Well, thank dog for the future, because in the space of a few Googles, I have determined that it's an idiom dating from the 1800s or earlier, which means (literally) to be frog-marched. In other words, one who is "walking Spanish" is the recipient of a frog-marching, and it describes the tiptoe progress of someone who is being hoisted up by the seat of his pants and propelled forward by pressure of a hand against the nape of his neck.
By extension, it also metaphorically described the progress of someone unwillingly ejected, or forced to carry out a task he does not prefer.
(And shall we even talk about the historical nationalistic hatreds revealed as linguistic artifacts by both "walking Spanish" and "frog-marching"?)
Oh, dear. I appear to have caught the tone of this New York Times article from 1877, describing the idiom and the practice, in the context of the forcible ejection from the halls of government of one Dr. Mary Edwards Walker, Congressional Medal of Honor recipient, for the offense of... wearing trousers.
Well, I guess it can only help me write Doc. And you can bet he's going to be using that idiom some point in the course of it.
*walks Spanish back to the book*