No. This is a 'meme' that has been circulating in various forms since pre-internet days.

It is mentioned and dismissed in Christian Wolmar- Fire and Steam.

It is easily disproved by actually measuring the width of wheel ruts in existing Roman roads- although of course this meme circulates mainly in the USA where practical refutation in this way is more difficult.

Different areas in the UK and mainland Europe standardised locally on different gauges for wooden wagonways ( which have had L shaped wooden rails to guide the wheels) in the late 18C. In the midlands of the UK this was usually 4foot 6inches, in Scotland about 5foot, in the North East of England about 4foot 9inches. The advantages of a local standard is that wagons were then able to be shared between wagonways, but, as there was no national traffic there was no benefit to a common standard.

We ended up with 4' 81/2" because the first Railway systems developed from wagonways in the early 19C in the North east of England and adding flanges to wheels about 4' 9" gave the railway gauge. George Stephenson started as a railway engineer in the north east and naturally designed railways with a gauge whch he knew pragmatically worked. He was hugely influential as the designer and builder of both the Stockton and Darlington and Liverpool and Manchester railways. After these were built it created a de facto standard ( although railways were built to other gauges until the issue was resolved by parliament in the UK.) The US ended up with this gauge because early equipment was imported from UK manufacturers.

I think it should be pointed out that gauges in the US varied widely for many years before interchanging became important. The adoption of 4'8.5" was mainly because that was the standard used by the Pennsylvania Railroad, then the largest in the world. It was the most important railroad for others to be able to interchange with.

Many railroads in the US had chosen a smaller gauge because it meant less land to purchase and pay taxes on. After the Civil War, the railroads in the south had agreed to rebuild on a 5' standard. I think the government in Canada had also adopted 5' as their original standard in the mid 1800s.

Most North American railroads converted to the current standard during the 1880s and 90s. For some it was an expense they couldn't handle and caused a few to go bankrupt or otherwise merge with other lines.