Here is an interesting chart showing the share of world stock markets at the end of 19th century and last year:

Click to enlarge

The size of the stock markets of France and the UK declined significantly in the period shown due to both the countries losing their status as world superpowers. However the U.S. became a major global player with its stock market rising from under 20% at the end of 1899 to about 45% of the world market size at the end of last year. From the Credit Suisse YearBook:

The New York Stock Exchange traces its origins back to 1792. At that time, the Dutch and UK stock markets were already early 200 and 100 years old, respectively. Thus, in just a little over 200 years, the USA has gone from zero to a 45% share of the world’s equity markets.

Source: Credit Suisse Global Investment Returns Yearbook 2012,  Credit Suisse