I wonder one thing about baseball. The teams get talent from Cuba, Canada and Japan. When is the first European gonna play in the Major Leagues, you think?
--Ken's cousin in Sweden
There have been around 150 European-born players (full list here) who have played in the majors. Nearly all of them grew up in the US. Four were born in Sweden, all of whom came over during the same immigration wave that brought our grandparents, Gottfrid and Helga Arneson, over to the US in the early 20th century.
The most famous home run in baseball history ("The Giants Win The Pennant!") was hit by a Scottish-born player named Bobby Thomson.
The best European-born player is probably Bert Blyleven, who many people think belongs in the Hall of Fame.
The list of players who actually grew up in Europe is very short. Here's a story about one of them I know of, Rikkert Faneyte from the Netherlands.
Getting a player who grew up in Europe to reach the Majors is kinda like getting a player who grew up in California to reach the NHL. The culture, resources, competition, and number of players just aren't good enough to produce very many good players.
There are some decent baseball leagues in Holland and Italy, but they're more likely to supply baseball players to US colleges at this point than to MLB. The quality of play in European baseball compared to the US is probably about where basketball was in the 1960s or early 1970s. You probably need a generation or two of Europeans coming to the US, playing at lower levels, returning with their knowledge and passing it on, before we'll see Europe as a legitimate source of MLB talent.
The nature of the game also makes it hard for someone from Europe to come over here and succeed. In basketball and American football, if you're big and strong and fast, there's probably a job for you. Baseball is a little more like soccer in that those traits are useless if you don't also have the talent and skills to play the game. It would be hard for a European athlete who spent his childhood playing soccer to come to the US as a young adult and succeed in baseball.
Hitting--deciding in a fraction of a second whether to swing or not, and then when and where to aim your swing--usually requires years and years of practice. You rarely see anybody switch sports to baseball at a late age and become a successful hitter.
It's more common to see someone switch over to pitching at a late age and have success. Being big and strong as a pitcher can help you throw hard, which is a big first step in succeeding as a pitcher. There have been some tall NBA players, for example, who went on to pitch in MLB.
I'd probably say it's much more likely that we'd see a European pitcher than a hitter, except for one thing: Europeans don't throw. Throwing is just not a part of European culture. The only reasonably popular sport played throughout Europe with any kind of throwing at all is team handball, and even then you rarely see European kids running around playing team handball on the playground. Every time I've seen a European try to throw a ball, it looks nearly as awkward as an American trying to throw with his opposite hand. I'd think that you need to do at least some reasonable amount of throwing as a kid, so that it's a somewhat natural motion for you, to have a chance to succeed as a pitcher as an adult.
When I lived in Sweden in 1988, I worked with a Canadian guy who helped coach a baseball team in Stockholm. When he heard I was a baseball fan, the first words out of his mouth were "Can you pitch?" He was obviously desperate. Looking back on it, I wish I had replied "depends on what you mean by 'can'", but I had never really pitched, nor had I ever heard of Bill Clinton.
There has been talk about MLB teams trying to get some cricket players to convert to baseball, particularly in Australia where both cricket and baseball are played, but it hasn't happened yet. Usually, those cricket players are making enough money playing cricket that it isn't worth the risk. But maybe there's some British kid out there playing cricket every day who will make his way over to the US and be the first person who grew up in Europe to become a baseball star.