Your categories (being the US census categories) are, of course, US-centric so cannot be directly compared to other countries. That is, in the Swedish context, why should Swedes and Iraqis both be categorized as "white" while Chileans are in their own category of "Hispanic or Latino"?
For some hard numbers, "15% of the [Swedish] population was born abroad, 5% of the population was born in Sweden to two parents born abroad, and another 7% was born in Sweden to one parent born abroad. Resulting in 27% of the Swedish population being of at least partly foreign descent." (Wikipedia for Immigration_to_Sweden#Current_population_of_immigrants_and_their_descendants . That also shows the complete breakdown, with Iraq, Bosnia, former Yugoslavia, and Iran in positions #2-5, after Finland.)
Compare that to the US: "According to the Census Bureau's 2009 American Community Survey, the US immigrant population was 38,517,234, or 12.5 percent of the total US population." (http://www.migrationinformation.org/feature/display.cfm?ID=8... )
12.5% for the US is less than 15% for Sweden.
It's hard to compare the Swedish immigration numbers to the ones in the link you gave, which talks about ethnicity in San Francisco, because the Swedish government tracks nationality, not ethnicity.
It's also difficult to compare San Francisco, which has a higher than average diversity in the US, with all of Sweden. For example, in Botkyrka Municipality near Stockholm nearly half the population has at least one parent born in another country, and that rises to 65% in one of its neighborhoods; a neighborhood where 26% of the neighborhood is a foreign national.
By comparison, that number is 30% (instead of 26%) for the SF Bay Area. However, even then it's not a good comparison because I picked a single neighborhood of 82k people in Sweden, against the Bay Area, which has a population of 7.15 million. The entirety of Sweden is only 9.5 million people.
So yes, you can easily say that San Francisco is a very culturally diverse city. No one disagrees with you.
However, purplelobster's point was that Sweden currently, and over the last two decades, takes in a lot of refugees on a per-capita basis, and even more than the US does - a fact which is not in dispute.