The lyrics of that song are ambiguous enough that it could be read that Sublime is just referencing past riots and Civil Rights demonstrations in many of those cities, going back to the 1940s, 50s, and 60s. Some of the cities may just be mentioned because they fit the song.
That said, there definitely were demonstrations nationwide after the Rodney King decision, and some of them did turn violent:
The most violent protests aside from L.A. were found in West Las Vegas (see page 57). One person was killed in a fire, and dozens of arrests were made in unrest that last a couple weeks.
Destructive demonstrations were also held in San Jose, where "rioting, window breaking, looting, fires and arrests" occurred over multiple days.
On May 1, 1992, the Los Angeles Times wrote of demonstrations in Seattle, Chicago, Miami, Washington, D.C., and Providence, Rhode Island. Also mentioned in the article are Denver and Phoenix, though it doesn't make it clear if there were actual demonstrations there or not, just that the police had been anticipating them.
"Urban violence was reported across America into the night Friday and wee hours Saturday," read a May 2, 1992, article by the news agency the United Press International. According to the article:
In New York City, "121 arrests" were made "from the time of the verdicts Wednesday to dawn Saturday. Forty-one people were injured, including 33 police officers, in incidents related to anger over the King case...Two cars were overturned not far from New York University and several restaurants were stormed in Greenwich Village...Businesses throughout the city...closed early."
In Atlanta, there were 40 arrests made during demonstrations.
In Springfield, Illinois, "a mob of 100 youths set small fires and hurled rocks and bottles at firefighters in a public housing project. Springfield police said 10 people were charged with mob action and criminal damage to property in the violent flareup that began around 12 a.m. and lasted four hours."
"30 to 50 youths broke several windows of a store" in Rochester, New York.
"Half a dozen fires broke out" in Chelsea, Massachusetts, outside Boston.
The article also reported some violence in Chicago, and a minor disturbance in Pittsburgh, while there were more peaceful demonstrations in Boston, MA; Buffalo, NY; Cambridge, MA; Richmond, VA; and Petersburg, VA.
According to an Associated Press article published on Saturday, May, 2, 1992:
Twenty-six people were injured during demonstrations in Atlanta.
Students from UC Berkeley blocked traffic on I-80 across the Bay Bridge to San Francisco. (According to the UPI article further up, the San Francisco Giants baseball game at Candlestick Park on Friday evening, May 1, was cancelled.)
A man was arrested in Cleveland "who disrupted a peaceful rally by twice driving by in a van while waving a flag showing a swastika" after driving the wrong way on a one-way street to get away from the crowd chasing after him.
In Madison, Wisconsin, 34 police squad cars had their windshields smashed in a parking garage with a note left at the scene which read: "Justice for King".
Students at UC San Diego burnt three effigies of police officers.
Also mentioned in the AP article are demonstrations in Minneapolis and Kansas City.
The New York Times reported on Sunday, May 3, 1992, that over the preceding days:
There were some non-violent protests held in New York City and New Jersey at the time, including a peaceful protest at City Hall attended by 500 to 1,000 demonstrators, where Al Sharpton addressed the crowd.
New Rochelle, in Westchester County, New York, experienced "vandalization and looting of 25 stores" which left 3 people injured.
"Six persons were arrested and five businesses were looted in the Journal Square area during a protest by 300 people" in Jersey City, New Jersey.
"27 people were arrested for blocking traffic during a demonstration" in Newark, New Jersey.
"70 high school students staged a mostly peaceful protest, but a few store windows were broken and three youths were arrested" in Bridgeport, Connecticut.
TL;DR: Yes, there were demonstrations across the United States, some of them with violence, though none of them were on the scale of what happened in Los Angeles at the same time.