In the January and February 1849 issues of his journal ‘Neue Rheinische Zeitung’, Karl Marx published articles calling for the extermination of whole races in Europe. These articles were included in a book of the teachings of Marx, Engels and Lenin published in Germany during 1902 and again in 1913. It is most likely that Engels wrote them. This socialist programme considered the Slav nations to be ‘counter-revolutionary’. The Germans, Poles and Magyars (Hungarians) were considered to be ‘the bearers of progress’. The rest must go:“The chief mission of all other races and peoples, large and small, is to perish in the revolutionary holocaust”.
It was explained that the Slavs had failed to pursue essential historic evolution, so were therefore counter-revolutionary. All European countries contain ‘left-overs of earlier inhabitants’, now rightly brought into subjugation by more advanced peoples. Amongst such ‘racial trash’ (Voekerabfall) were listed Scottish Highlanders, Bretons, Basques, South Slavs (Slovenes, Croats, Serbs) and Czechs.“Until its complete extermination or loss of national status, this racial trash always becomes the most fanatical bearer there is of counter-revolution, and it remains that. That is because its entire existence is nothing more than a protest against a great historical revolution. … The next world war will cause not only reactionary classes and dynasties, but also entire reactionary peoples, to disappear from the earth. And that too is progress”.
As Marx and Engels aged, they took greater interest in Eugenics and Social Darwinism suggesting that progress was interpretable in racial terms.
For what I found, the article to which Watson is referring to is titled ‘The Magyar Struggle’, and was published in number 194 of the ‘Neue Rheinische Zeitung’ journal on the 13 of January 1849 (which can be found in full at: http://www.marxists.org/archive/marx/works/1849/01/13.htm). By reading this article you can clearly see that the main problem with Watson's theory, is that Marx did not write that article, Engels did; a huge mistake for an unbiased literary historian (though he latter admits the possibility that Engels, and not Marx, wrote it).
Then we have Watson’s claim that Marx (Engels in any case) called for the extermination of whole races in Europe. Of course, if we pick and chose phrases from the article and take them out of context, then it does look like Engels (not Marx) wanted to get rid of other ethnic minorities (and not races). But if we read the whole article, it seems clear that Engels was talking about them in the context of the European Revolution of 1848, on the role that some ethnic minorities played in previous conflicts (as fanatical standard-bearers of counter-revolution and supporters of European monarchies), and on them being taken into account in any future revolution; and not because they were from an ethnic minority or from a different race (as Watson suggests).
Then we also have the issue of the "Völkerabfälle", which by the way does not mean “racial trash”, it literally translates as “residual fragments of people” (and this is how it's translated in the English version of the “The Magyar Struggle” article). This term was not used by Marx (as he didn’t write the article in the first place), and it was neither from Engels; it was from Hegel, as Engels explains in the article.
Therefore, it looks to me like the answer to the question on the title of this post is: Absolutely not. This rumour is no more than false anti-communist propaganda and a deliberate and malicious attempt to demonise Karl Mark. As Latvian political scientist and cultural commentator Ivars Ījabs explained: “To present Karl Marx as the progenitor of modern genocide is simply to lie”.