The assassination and surrounding events certainly led to the passage of the Act. Your question is whether or not the riots in particular had an effect, which is a bit more difficult.

Senator Charles McC. Mathias, who supported the Act, writes that the "crisis in race relations" led to federal action. This quote, taken from his report, summarizes the history as the Act's passage as its Congressional supporters understood it:

On the night of April 4, Dr. Martin Luther King, Jr. was struck down by an assassin’s bullet in Memphis. Another series of civil disturbances followed, including one in Washington, D.C., that required the President to call out the National Guard and impose a night-time curfew. The crisis in race relations in our country forced Congress to come to grips with these tensions. The Rules Committee, jolted by the repeated civil disturbances virtually outside its door, finally ended its hearings on April 8. The next day, it reported to the full House a rule for debate that agreed to the Senate amendments, including the compromise fair housing title, and prohibited any additional amendments. The following day, April 10, the House debated for one hour the Civil Rights Act of 1968 and passed it 250–71. The very next day, President Johnson signed the bill into law. (bolding mine)

On April 5, President Johnson sent this letter to the House and to reporters urging the passage of the act, writing that "the Nation so urgently needs the healing balm of unity." Prior to the riots, the King assassination itself had rekindled the debate over the Act, but it seems that the "civil disturbances" really affected Congress.

So, were riots effective? I read this question as standing in for a larger counterfactual: "had people instead demonstrated without property damage, would they have gotten the same results?" I don't know if we can give a definitive answer, but I'll say that Matthias's account comes about as close to a direct statement that the government succumbed to the pressure of riots as one could expect.