Should the justices hear the case and again rule against the state, Herbert said the state will give full recognition to same-sex marriages.
"Once the law is the law of the land, my job is to respect it and enforce it," Herbert said.
The governor downplayed the potential expense of going to the Supreme Court. He said the anticipated cost has already been built into state budgets. He also expressed support for Gene Schaerr, the attorney hired by the state to lead Utah's defense of its marriage ban.
Herbert said his own opinion about same-sex marriage is irrelevant to his belief that the appeals should continue, rather it is his belief that a resolution is in the best interest of all sides. If Utah's law had been written "to allow same-sex marriage, I'd be defending that law," he said.
The governor would not say if Utah would seek a review of Wednesday's decision by a full panel of the 10th Circuit, or if the state would just ask the Supreme Court to take the case. He just wants a resolution to the issue.
"This is a significantly important issue, landmark probably in its outcome," Herbert said. "[Same-sex marriage] is not like typical lawsuits out there."
In other issues, Herbert said he still hopes to reach an agreement with the Obama Administration on his "Healthy Utah" plan for expanding health coverage to low-income Utahns. Herbert said he believes 111,000 poor Utahns can get coverage entirely with federal money and potentially even save dollars.
If, after three years, the program is not working or employers are found to be dropping coverage and moving workers to the state system, then a "fallback" position is possible, where benefits could be dropped for those making between 100 and 133 percent of the poverty level.