The same survey that had that surprising result found health care is the top issue cited by voters going into the 2018 midterm elections.
But it ranks much lower among voters who live in areas that will have competitive congressional or gubernatorial races, according to the Kaiser Health Tracking Poll.
And the same poll found that most people were unaware of the fact that, beginning in 2019, the Obamacare mandate that currently requires most Americans to have health insurance will be effectively repealed.
Just 36 percent of voters knew Congress had suspended the tax penalty for not having insurance as of next year. And 46 percent incorrectly said it had not been repealed, with the remainder being unsure.
Kaiser's poll was conducted last week, in advance of and in the midst of a three-day shutdown of the federal government.
At the same time, some Republican legislators are pushing for a renewed effort to repeal and replace key parts of Obamacare — while some of their GOP counterparts want to pass legislation that could help stabilize marketplaces that sell Obamacare plans.
On the heels of the shutdown, Senate Minority Leader Chuck Schumer, D-N.Y., on Tuesday said Democrats
Trump, in turn, lashed out at Schumer, saying that if he didn't get funding for the wall there would be no deal on the Deferred Action for Childhood Arrivals program, which for now has allowed a group of immigrants known as "dreamers" to remain in the United States.
But Republican voters may be less concerned about the wall than the president, according to Kaiser's poll.
The poll found that 43 percent of GOP voters said that passing federal funding to build a border wall should be a major priority for Trump and Congress.
That was just 1 percentage point higher than the Republican voters who said stabilizing the Obamacare marketplace was a majority priority.
The leading priority for GOP voters, cited by 48 percent of them, was repealing Obamacare. Another 43 percent said that addressing the prescription painkiller epidemic was a major priority.
Among all voters, 29 percent said health care was a key issue for candidates to talk about going into the 2018 elections for Congress and governorships.
Another 27 percent said "the economy and jobs" were a key issue, followed by 24 percent who mentioned immigration, and another 24 percent who cited the situation with North Korea.
However, health care was less important in so-called battleground states, which are having competitive elections in 2018.
In such states, 34 percent of voters gave the economy and jobs as a key issue, while 23 percent said the North Korean situation was a key issue. And 22 percent said immigration was a key issue.
Health care was named by 21 percent of voters as a key issue, just 3 percentage points ahead of the federal budget deficit.
Obamacare advocates hope to make health care a deciding issue in this year's elections and have pointed to polling for elections held in recent months as evidence that voters are motivated by that issue.
The Public Policy Polling survey in November found that 43 percent of the voters in Virginia's election for governor, which was won by Democratic candidate Ralph Northam, said health care was "very important" in their decision about whom to vote for.
Brad Woodhouse, head of the Obamacare advocacy group Protect Our Care Campaign, told CNBC earlier this month that it will use findings of another poll that rated health care a top priority among voters to push Democrats and progressives to campaign on that issue.
"The Republicans have done what we never could," Woodhouse said.
"They have made the Affordable Care Act popular, and they have made it a political anvil around their necks" by trying to repeal it, he added.
But Kaiser's poll on Friday suggests that health care could be less of a deciding issue than Woodhouse hopes, at least in battleground states.
However on Thursday, Protect Our Care said it sponsored a ''simulation'' that surveyed voters across 18 battleground congressional districts where the incumbent voted for the recent GOP tax overhaul and also voted for legislation to repeal and replace the Affordable Care Act.
''Aggregating the data from all 18 districts, the polling memo shows that voters learning about their Republican Member of Congress's actions on health care or taxes had lower approval relative to the control group by statistically significant levels. The simulation gives conclusive proof that the more voters hear about these two Republican-led plans, the more voters will [reject] the Republican proponents for doing it," the group said.
Kaiser's survey questioned 1,215 adults from Jan. 16 through last Sunday. It has a margin of error of 3 percentage points.