Can you explain what you mean by "socially unacceptable?"

Horse-drawn vehicles are still commonly seen in Amish and Mennonite areas, and were not uncommon throughout the South well into the 1960s. The only highways where they cannot be used, even today, are freeways.

I think it’s going to be hard to pin down an exact date or date range - as /u/mrdowntown points out, horses are still used on public roads in some communities, and I don’t think there is a definite line where horse use plateaus to its current state. But I will try to sketch out some rough parameters.

Certainly, Automobile partisans from the very beginning styled the new technology as a replacement for the horse (as a means of personal transportation, anyways), and by at least 1910 many of those partisans considered horse traction to be obsolete. By the early 1910s, trade journals like Horseless Age, MoToR, and others were definitely publishing letters and articles casting horse traction as an obsolete (if not extinct) form of transportation - often in the form of complaints about horses’ interference with their ability to operate automobiles as they pleased.

Of course, market penetration of the automobile was far from complete in the early 1910s; sales of automobiles to farmers really only began in earnest in 1906, and prior to that there was considerable (sometimes violent) rural opposition to cars. Once the economic (and social) benefits of automobility became apparent, and once cars came within the financial reach of farmers (through considerable price reduction and later the introduction of payment plans), rural Americans over the course of the 1910s and 1920s quite rapidly adopted the automobile (check my post here for more information and hard figures), led first by “high wheeler” cars and then by the ubiquitous Ford T. Even for those who were not convinced by economic benefits (which were frequently touted in advertisements), once there reached a critical mass of automobility horse traction became more and more marginalized. Road speeds increased (although less significantly in rural areas), horses natural aversion to automobiles created new dangers, and drivers grew less and less patient with horse traffic on their roads. I don’t think it is a stretch to say that by the 1930s at least, anyone regularly using horses on public roads was doing so either out of religious conviction, because they couldn’t afford an auto, or due to personal resistance to the technology.

While "socially acceptable" is a hard thing to pin down from an historical perspective, I think a good case could be made to answer this question with, "By the end of the 1920s," because in the 1930s, the phrase "horse and buggy days" began to be used as political fuel.

First, a couple milestones: In New York City, the last horse-drawn streetcar (or "horse-car") closed down in 1917. In Pittsburgh, the last horse-drawn streetcar closed down in 1923.

But the question at hand asked about "rural roads". As one might expect, the "horse and buggy" era lasted a little longer in the country.

In 1928, about 4.6 million automobiles were sold in the United States. The U.S. was industrializing quickly, as exemplified by an ad from the Peekskill National Bank in the local newspaper The Highland Democrat that ran on September 1, 1928:

"Not So Many Years Ago the Jingling, Jogging Horse Car, Rumbling Along As Far As the City Limits, Was a Great Convenience. Today We Step Into Our 60-Miles-An-Hour Motor Cars For Trips of a Mile—Or a Thousand.

"RADICAL changes have come, too, in the handling of money. In horse car days folks carried most of their money with them, or hid it in supposedly secret places. The modern way is to put money in an account with a financial institution such as this—for safe-keeping, for convenience, for profit, for credit and money—independence building.

"Which method are you using—"horse car" or "motor car?""

But then, the Great Depression hit. And that set industrialization back for a short while. Gasoline became expensive and unaffordable for many lower class people, which led to something derogatorily called "Hoover carts"--a functional automobile being pulled by horses because the owner couldn't afford to put gas in their car anymore.

The Democrats did not soon let the Republicans forget about this. In 1935, when FDR was trying to push his Supreme Court-packing idea, he gave an impromptu speech where he accused the Republicans of living in the "horse and buggy days" (1, 2). A song was even written about it.

In 1948, Harry Truman gave a campaign speech reminding voters how bad things had been under the Republicans and how "Hoover carts" had become so prevalent in the early 1930s, and how that had ended under the Democrats.

And in 1952, supporters of Adlai Stevenson's presidential campaign once again brought back the stark image of the "Hoover carts" in a now-industrial age just twenty years removed from Herbert Hoover's presidency.

The "horse and buggy age" finally did seem to come to an end by the end of the 1930s. In 1938, the New York Times reviewed a book called The Horse and Buggy Doctor, a nostalgic memoir written by an aging rural doctor. The review began:

"The country doctor of horse and buggy, or saddlebag, days is more or less a vanished figure. The modern medicos drive fast cars, one man can serve an extensive district, and, more important than his own transportation perhaps, an automobile will get the patient to a hospital if necessary."

On June 1, 1939, the rural (or, at least, exurban) Peekskill newspaper The Highland Democrat ran an article referencing horse-drawn transportation as a mode that was nearly dead:

"'Get a horse,' the populace shouted when one of those hardy pioneers of 'gas buggy' days chugged down the dusty streets of 30 and more summers ago.

"The older heads scoffed at this noisy innovation in travel, cheerfully predicted it would never be more than a toy. Today the automobiles have so crowded the highways that current wisecrack centers around a horse being frightened, not by an automobile, but by the sight of another horse..."

In Europe, the era may have gone on a little longer. The Highland Democrat ran a blurb in 1928 about the town of Bois-le-Duo, Netherlands, where they noted that the horse-car was "still in fading competition with the Chevrolet bus," unlike in the U.S. where public transportation was universally mechanized.

But by the end of the 1930s, horse transportation seemed to be a thing of the past in Europe, too. The The Agricultural History Review ran a 1963 article entitled "The Development of Mechanization in English Farming", which gave a list of equipment of a 200-acre farm in Yorkshire in both 1937 and 1939. In 1937, this farm was still using horsepower for much of the work, but the 1939 list shows that the horses had been replaced by tractors, and the farm had entirely mechanized.

TL;DR: Depending on how you define "socially acceptable," you could probably say that, by the end of the 1920s, in the United States, the more successful rural families had replaced their animal transportation with automobiles, and those families who hadn't were working toward that goal. The Great Depression set that goal back for a few more years for families hit hard by the Depression, and "Hoover carts" were used as a political attack. By the end of the 1930s, the "horse and buggy days" were looked at as a thing of the past and having an automobile was the norm. In Europe, this may have gone on a little longer, but by the time World War II broke out, automobiles had largely replaced animal transportation rural areas there, too.