Rights focus on general principles. You have a right to speak, but no right to a printing press or blogging software. You have a right to bear arms, but no right to be provided with a firearm. You have a right to exist, and to control what happens to your body, but no right to be provided with a doctor. A right cannot impose a negative cost on others.
What he means in the first case, I think, is that the right to speak doesn't confer a right to be heard. To be more accurate, he should have said that we have no right to computers, since once you have one, and you get internet access, blogging software is generally free. By saying that we have "no right to be provided with a firearm," he's not saying that the government can prevent us from having them, but that the right to opportunity does not guarantee you the possession of the desired object. For Welch, the right to bear arms means only that the government can't infringe on your right to have arms if you can get them.
Guaranteeing things that society considers necessary goods violates natural right as far as Welch is concerned because it compels people to provide the goods without what Welch might consider appropriate compensation. "Socialized health care, in any form, requires others to suffer financially for it to exist," he writes, "A right also has to exist on its own; it cannot require that another right, or the rights of others, be violated." The right violated, presumably, is that of the owner of resources to demand or negotiate for fair compensation. Affirming a right to health care as a guarantee makes it unconditional that the owner of resources must provide them, whether he wants to or not and regardless of whether whatever compensation he receives seems fair to him.
Welch's fallacy, of course, is the assumption that natural rights are the only rights that can exist. His principles effectively limit the effectiveness of civilization because the rights of individuals, based on the presumptive state of nature, would always trump whatever even the most democratic society might choose to define as a common good. He seems incapable of envisioning circumstances in which people might trade in their natural rights, so to speak, in return for guarantees that might make society less brutally competitive. He most likely takes seriously Jefferson's declaration that certain rights with which men are "endowed by their creator" are inalienable. Men either cannot or should not surrender the right to pursue happiness to the material maximum, for instance, in favor of a more secure social life. In more complacent terms, political society requires no sacrifice from the self-reliant in Welch's scheme; they must give up nothing, and especially not their right to decide for themselves how much they deserve, for the sake of social peace.
Refuting Welch requires no elaborate argument. All it takes is to affirm that people can confer substantive rights upon themselves in the course of creating a political society. Of course, no such affirmation can ever guarantee absolutely that we'll get what we claim as rights, but it's no answer to the claim that making it is wrong because it violates the rights that someone claimed for himself prior to entering into society, or that society can only confer the rights that already adhere to man in the state of nature. If that were true, the whole procedure would be redundant. The difference between society and the state of nature, presuming for entertainment purposes only that there are such things as natural rights, is that rights conferred by society come with obligations on its members. Citizenship comes with an obligation to provide for the common defense, promote the general welfare, etc., according to society's definition of these goods. People who refuse to accept an obligation to contribute to the general welfare by sharing resources may as well go live in the woods, since they see the state as nothing more than a police force to protect them from the have-nots of society while they accumulate wealth without limit.
Welch might claim that his views are more consistent with the Founders and Framers, but that depends on more information than we get from his letter. I'd readily concede that few if any Founders would have assumed a right to health care, but we do know that they were suspicious of the unlimited accumulation of wealth that passes for "freedom" for too many modern Americans. They believed that excessive wealth led to luxury and an inevitable corruption of society and politics. They depended, perhaps naively, on notions of virtuous frugality and simplicity to restrain greed. I wonder sometimes whether they'd reject as completely as some suspect a scheme to check wealth by diverting it to unimpeachable social goods like health care. We can guess, but we can never know the answer. But they left us a mechanism that empowers the people to figure such matters out for themselves and shape policy accordingly, and "natural rights" have no veto on that process. They're fine for purposes of intellectual argument, but anyone who thinks they're the last word on any subject has a lot of thinking yet to do.