(CNSNews.com) - "We all saw what happened on the Texas border last summer, but we need to understand that the problem is not going away," Texas Gov. Greg Abbott told CBS's "Face the Nation" with Bob Schieffer on Sunday.
"Already this calendar year, since January 1, we have had more than 20,000 people come across the border, apprehended, unauthorized. And so we have an ongoing problem on the border that Congress must step up and solve."
(All told in fiscal year 2014, U.S immigration officials removed 213,719 individuals apprehended while attempting to unlawfully enter the United States.)
Schieffer reminded Abbott that there are some 800,000 illegal aliens living in Texas right now. "You don't have enough buses to send them back to Mexico, and I don't expect you can put all of them in jail. What are you going to do with them?" he asked Abbott.
"Well, two things," the governor responded. "One is, the president himself said as these people were coming across the border that he would repatriate them as soon as possible. So, we need to see whether or not the president himself is going to live up to the commitment that he made."
Abbott said the second thing is to let Congress -- not the president -- decide how to deal with the millions of people who are in the United States illegally: "And so we need Congress to have the latitude to fulfill its responsibility to solve the problem."
Abbott said as governor, he has outlined a plan to secure the border: "I'm going to add more than 500 more Department of Public Safety officers, more Texas Rangers, more technology. We are coming out of our own pocket, Texas taxpayers' pockets, to secure the border and doing the job that the federal government must do.
"The first step that has to be taken in this whole process is to secure the border," he added.
Abbott filed a lawsuit on behalf of 26 states challenging President Obama's decision to go around Congress by unilaterally giving millions of illegal immigrants permission to temporarily stay in the U.S. and get Social Security numbers so they can work here as well.
A federal judge recently put Obama's plan on hold, and Abbott said he expects the case to "continue through the court process all the way to the Supreme Court."
Abbott said it is "essential" to win on constitutional grounds, "because what we have here is a situation where the president has violated the rule of law and really contradicted the Constitution by actually making up the law and imposing his own standards on the immigration system.
"And so what Congress must do is to continue to ensure that the rule of law and the United States Constitution as constructed are fulfilled. And that means it's the Congress, not the president, that establishes what our immigration laws are."
Abbott noted that the lawsuit he filed doesn't deal with immigration -- it deals with "an overreach by the president."
In announcing his deferred deportation plan for millions of illegal aliens last November, President Obama said, "If you plan to enter the U.S. illegally, your chances of getting caught and sent back just went up." He said the new policy does not apply to recent border-crossers.