The senator even went as far to say that Mr. Obama's deliberations might have the effect of "emboldening our enemies."
Cornyn spoke with John Dickerson, host of CBSNews.com's "Washington Unplugged," on "Face the Nation" Sunday.
The senator's comments contrasted sharply with those of Sen. John Kerry, the Massachusetts Democrat who chairs the Senate Foreign Relations Committee. Kerry whether to approve a request from Gen. Stanley McChrystal for more troops there.
"I don't think our national security should depend solely on that," Cornyn told Dickerson. "But clearly that's an important part of it. I hope President Karzai understands that our national security interests don't depend entirely on his decision there whether to allow a recount. Obviously the legitimacy of that government is an important component of it. My point is it shouldn't be the lynch pin for us deciding whether to protect our national security interests in that region."
CBSNews.com Special Report: Afghanistan
Cornyn mentioned the strategy sessions Mr. Obama has recently held with his Afghan policy advisers, noting "deliberation is a good thing when it comes to fighting wars," but discouraged the administration from deliberating for much longer.
"At some point deliberation begins to look more like indecisiveness which then becomes a way of emboldening our enemies … and causing our allies to question our resolve," Cornyn told Dickerson. "We shouldn't let one component of this determine our national security here, which depends on providing an Afghanistan which denies a safe haven to terrorists, as well as stabilizing Pakistan. Those are our two national security interests at stake in Afghanistan."
Cornyn, who heads the National Republican Senatorial Committee, talked more about concerns he had about the administration's handling of national security threats abroad.
"You have to look at Afghanistan also in a global context where we've canceled basically our missile defense system, undercutting the Czech Republic in Poland," Cornyn told Dickerson. "We've I think not dealt with Iran with the kind of resolve that would show that we understand the nature of that threat. I think all of these are data points that begin to create a narrative or begin to create a picture that shows a lack of resolve when it comes to our national security."
Dickerson noted that Cornyn sits on the Senate Finance Committee, which last week voted on its bill to reform the nation's health care system. All of the committee's 13 Democrats and Maine Republican Sen. Olympia Snowe voted for the bill. Cornyn and the committee's eight other Republicans voted against the bill.
CBSNews.com Special Report: Health Care Reform
"Do you think Republicans will support the final bill and will there be more than just Olympia Snowe?" Dickerson asked.
"John, we're still waiting for the president's plan," Cornyn said. "So far we've been largely shut out of the process. Almost every amendment we've offered to try to improve the proposals has been voted down along party lines, and what we see now is the Democrats who hold the filibuster-proof majority - 60 votes - they're finding differences between themselves, but I think clearly we could find a way to work on this on a bipartisan basis. So far the president has let Democratic leaders in Congress basically run the show to the exclusion of any constructive Republican suggestions"
Dickerson highlighted a recent finding in a CBS News poll that .
"Are you worried about being in the category where you only have 29 percent on this question of health care?" Dickerson asked.
"I think they want something good to pass," Cornyn said. "If you ask more detailed questions, some of the components of the president's proposal - or I should say the leadership's proposal - in the Senate and the House are very unpopular."