Occasionally I see questions about whether the constitution should apply to civil lawsuits in which the government is not a party -- for instance, most libel lawsuits, child custody decisions, and the like. New York Times Co. v. Sullivan (1964), as well as other First Amendment cases, have famously said "yes," because even in civil lawsuits a government actor (the judge, jury, or both) is applying a government-created rule of law using government coercive power. And I think this is logically right: Imagine, for instance, that courts or the legislature adopted a rule that anyone offended by racist, sexist, antiveteran, or blasphemous speech could sue the speaker. Whatever one thinks the First Amendment result ought to be (and there is a hot debate, especially as to similar, though narrower, rules in workplace harassment law), surely there needs to be a substantive First Amendment analysis. The fact that the speaker must pay civil damages (compensatory and maybe even punitives) shouldn't distinguish the case from the government's imposing monetary fines for such speech.
(Contract disputes, such as those in Shelley v. Kraemer, are a different story; Shelley is rightly more controversial than New York Times Co. v. Sullivan, and has been less productive of further precedents. But the short answer, I think, is that a breach of contract or trespass to property lawsuit does involve government action, though in many situations may not involve unconstitutional government action, see, e.g., Cohen v. Cowles Media Co..)
But people still ask: Is this consistent with the original understanding of the First Amendment (or perhaps of the Fourteenth Amendment, when state litigation is involved)? This may be largely irrelevant as a practical matter, given how much precedent endorses the Sullivan view. Even many judges who generally approve of originalism are unlikely to disturb such solid bodies of precedent; Justice Scalia, for instance, has expressly said this in other contexts, and while Justice Thomas is more willing to reverse well-established precedents, even he is unlikely to go to the original meaning in all contexts, and when he is, he's often alone.
Here's my answer, based on reading a bunch of early free speech/press cases over the years: Free speech and free press objections to civil libel and slander lawsuits were often raised in the early decades of the Republic, and while they were often rejected on substantive grounds -- the courts took the view that defamatory speech was constitutionally unprotected, whether against civil liability or criminal punishment -- they were not rejected on "lack of government action" grounds. Sullivan's insight that even civil litigation still involves government power and is thus subject to government constraints seems to have been broadly accepted in the late 1700s and early 1800s. See, e.g., Root v. King, 7 Cow. 613 (N.Y. Supp. 1827); Mayrant v. Richardson, 10 S.C.L. 347 (S.C. Const. Ct. App. 1818); Reid v. Delorme, 4 S.C.L. 76 (S.C. Const. Ct. App. 1806); Harris v. Huntington, 2 Tyl. 129 (Vt. 1802); Middlesex Gaz., Mar. 12, 1791, at 1 (reprinting charge to a jury in Freeman v. Gardiner); Am. Mercury, Dec. 26, 1799, at 2 (charge to the jury in Rush v. Cobbett). And in Mayrant, Reid, and Harris (the latter two of which were cases involving the right to petition the government, rather than just the freedom of speech or press more generally), the plaintiff actually prevailed, based partly on constitutional principles.
So the Framers recognized that courts, including courts applying court-made common law rules, were agents of the government and generally subject to constitutional constraints. The early decisions were all state decisions applying state constitutions; but my readings suggest to me that -- despite the use of the word "Congress" in the First Amendment -- the scope of state freedom of speech/press/petition guarantees was seen as largely interchangeable with each other, and with the First Amendment. (Certainly early federal court cases, even ones applying judge-made speech restrictions and not statutorily enacted ones, treated the First Amendment as at least potentially applicable.) So my sense is that the original understanding of the First Amendment, and also the traditional one in the centuries since the Framing, is that it does apply to civil litigation, though the exact scope of the constitutional rules has of course changed over time.