A simple examination of history should put this fallacy to rest.
Automation has been replacing manual labor for as long as manual labor to accomplish a specific task has been performed, at least several thousand years. And most of us still have jobs, despite the automation and a growing population. How did that happen, despite all the predictions of gloom and doom?
Probably the first bit of automation was the animal drawn plow. One person with an ox drawn plow could cultivate the same amount as many people with hand tools. Did that put the many people out of work? No, it did not - now that everyone wasn't engaged in food production, some could specialize in more advanced fields, like building better plows, breaking oxen to harness, building and operating wagons to transport all that extra food, or what have you.
Moving forward, consider the case of Ned Ludd, a weaver in late 1700's England, who smashed up some automated knitting machines. Granted, Ludd appears to have been a ne'er do well who was championed for the wrong reason, but the coming of the automated loom as opposed to hand spinning yarn did not put everyone out of work.
The automated loom dropped the price of clothing to the point where more people bought clothes more frequently, creating more jobs at the automated factory, plus more jobs to meet the increased demand for wool and cotton, plus more jobs to transport the raw goods and finished clothing into the new stores with new jobs to sell the clothing to meet the increased demand...
Eli Whitney automated two manual processes: extraction of cotton (the cotton gin) and a standardized firearm production line (previously, guns had been pretty much handmade). Both reduced the number of people necessary for the task, neither ended up putting people out of work. Drop the price of the item being produced, demand goes way up, production goes up, more jobs, and more jobs to supply those factories, and to transport the goods.
Arguably, the coming of the steam railroad put a lot of horse drawn wagon operators out of a job. But, the steam railroad so boosted commerce in general that there were plenty of jobs to go around.
And, one of the most ironic cases of automation and job loss: Henry Ford's auto production line. Ford automated much of the process of building an automobile, previously done by hand. Ford dropped the price of a car by an order of magnitude, by lowering labor costs. Did this eliminate jobs? In fact, it did the complete opposite. Ford's pay of $5/day for work on his automated assembly lines essentially created the middle class, and in the process, Ford created a lot of customers... for his cars. More cars sold meant more demand for metal (more mining/smelting jobs), more demand for gasoline (jobs drilling, refining, and selling gas), more need for repair (jobs repairing automobiles)... it just kept growing. The cheap automobile made a lot of new jobs possible that weren't efficient before, like traveling salesman, delivery person, etc...
Finally, let's look at a contemporary situation. E-commerce. It is putting some brick and mortar stores out of business. Oh, dear, that's automation killing jobs, right? In fact, the opposite has happened. E-commerce relies on delivery, with UPS and FedEx seeing explosive growth in delivery. More jobs. The websites must be maintained - being a good web developer today is a fairly lucrative occupation. E-commerce with its lower prices and greater selection has increased sales, which increases production, which increases jobs elsewhere.
For as long as civilization has existed, automation has been eliminating jobs. If you look at previous examples, not by today's standards, but within the context of their time, you should be able to see that for every job that automation eliminates, the results of the automation creates more than one job.
Automation is not only not detrimental to employment, it is mandatory for increased employment.