In response, self-described "supporters of the 2nd Amendment" and Republican candidates for president have opposed his actions, saying that they are "unconstitutional" and should be repealed; however, a look at what the actions contain shows that the 2nd Amendment is not under threat, despite what a vocal minority seems to think. Here are some of the provisions:
"Background checks for all gun sellers, overturning current exemptions to some online and gun show sellers"
This seems entirely reasonable and in line with what the public wants. A Pew Research Center poll from last August found that 85% of Americans, including 79% of Republicans, support greater background checks for weapon sales.
"Increased workforce for the FBI to process background checks, hiring more than 230 new examiners"
This means that any backlogs relating to gun backgrounds checks will be cleared more quickly because of the increased staff. Theoretically, people waiting to buy a gun might be able to purchase them quicker.
"Congress being asked to invest $500m (£339m) to improve access to mental healthcare in the US"
The aforementioned poll notes that Americans support limiting weapon sales to mentally ill people; in addition, many of those against gun control measures are also in favor of such a limit. Asking Congress to improve mental healthcare seems like positive step to make; I don't see why anyone would be against this.
At any rate, no president can unilaterally nullify an amendment; it requires the states themselves and Congress to perform such an action. Firearms are not being banned - if President Obama really were the "dictator" his opponents make him out to be, he would already have banned them by now. He wouldn't be bothering with executive orders that are short-term fixes.
It is entirely possible to be pro-2nd Amendment while also supporting measures to exercise caution during gun sales. For example, the poll notes that 79% of Republicans support stronger background checks; this is significant because Republicans are generally more hostile to gun control than Democrats. From personal experience, most of the gun owners I knew in the United States supported background checks on the sales and were distrusting of buying weapons from unofficial venues.
Finally, regarding any such actions or legislation pertaining to firearms sales being "unconstitutional", it should be noted that at least three amendments (the 15th, 19th and the 26th, in particular) all defend voting rights. Using consistent reasoning, any restriction on voting rights is also unconstitutional and any state with voter ID laws or other methods to limit voting is going against the Constitution.