Texas, a state well-known for its close ties to the oil industry, has been able to mitigate the economic damage done by crude's price decline by cutting taxes and relying on its diversified economy, Gov. Greg Abbott said Tuesday.

"While other states have been raising taxes, we cut taxes by $4 billion ... and it is attracting businesses from New York, from New Jersey and across the entire country," Abbott said in an interview with CNBC's "Squawk on the Street."

U.S. crude prices have declined nearly 45 percent year over year and more than 8 percent in 2015. In fact, WTI futures dropped as much as 2 percent Tuesday on news Iran and world powers led by the U.S. had agreed on a historic nuclear deal before rebounding to move higher.

Eddie Seal | Bloomberg | Getty Images

Floor hands make a pipe connection on the drill string on the Orion Perseus drilling rig near Encinal in Webb County, Texas.

"Texas leads the nation in technology exports, we have the largest medical center, not just in the United States, but in the entire world," Abbott added, also saying that General Motors is currently investing $1.4 billion to open a new manufacturing facility in Arlington.

Nevertheless, while Texas' economy is large and very diversified, the state's revenue could take a hit from lower oil prices, according to an April report from the Dallas Federal Reserve Bank.

"Texas, owing to size and diversification, obtains 9 percent of tax revenue from oil and gas. Oil and gas severance tax revenue in Texas totaled $4.5 billion in 2013. With the halving of oil prices, and potentially lower production, those receipts likely will significantly fall in 2015," the report said.

Nevertheless, Texas' exposure to the oil glut is much smaller than that of other states, the report added, as crude accounts for 80 percent of Alaska's tax revenue and nearly 50 percent of North Dakota's.