As a specific form, prose poetry is generally assumed to have originated in 19th-century France as a reaction against dependence upon traditional uses of line in verse.That is certainly the truth.
At the time of the prose poem's emergence, French poetry was dominated by the Alexandrine, an extremely strict and demanding form that poets such as Aloysius Bertrand and Charles Baudelaire rebelled against. Further proponents of the prose poem included other French poets such as Arthur Rimbaud and Stéphane Mallarmé.
The prose poem continued to be written in France into the 20th century by such writers as Max Jacob and Francis Ponge.
At the end of the 19th century, British Decadent movement poets such as Oscar Wilde picked up the form because of its already subversive association. This may have hindered the dissemination of the form into English because many associated the Decadents with homosexuality; hence any form used by the Decadents was suspect.
Notable Modernist poet T. S. Eliot wrote vehemently against prose poems, though he did try his hand at one or two. He also added to the debate about what defines the genre, saying in his introduction to Djuna Barnes' highly poeticized 1936 novel Nightwood that this work may not be classed as "poetic prose" as it did not have the rhythm or "musical pattern" of verse.
In contrast, a couple of other Modernist authors wrote prose poetry consistently, including Gertrude Stein and Sherwood Anderson. In actuality, Anderson considered his work to be short fictions—in the current term, "flash fiction". The distinction between flash fiction and prose poetry is at times very thin, almost indiscernible.
There seems to be two rules of writing a "prose poem":
1. Random carriage returns.
2. Declaring it a poem.
Here is the first chunk of Billy Collins' "Table Talk" in this week's New Yorker:
*****
Not long after we sat down to dinner
at a long table in a restaurant in Chicago
and were deeply engrossed in the heavy menus,
one of us - a bearded man with a colorful tie -
asked if any one of us had ever considered
applying the paradoxes of Zeno to the martyrdom of St. Sebastian.
*****
When there are no longer any rules about an art form, the rules for successful art becomes all about your personal clout, not about the work itself.
Art without rules perfectly caters to the strengths of the upper classes and the already-famous, who tend to have lots of personal clout. If THEY say it's poetry, then it is. Because they make the rules, not the art form itself. Art without rules creates a power vacuum that the socially powerful will immediately fill.
Prose poetry is about taking back the power of the carriage return from the fascism of a well-formed paragraph. How brave.
I call bullshit on it.
And it's not even an exciting new innovation - it's been around for over a hundred years.