Any pregnant woman can travel to the US and give birth to a US citizen because of the 14th Amendment to the United States Constitution. The reason that the 14th Amendment exists is because of slavery.

Let's rewind a bit. Four years before the South seceded, the Supreme Court decided Dred Scott v. Sandford, 60 U.S. 393 (1857). In Dred Scott, a black slave named Dred Scott had been brought into the Wisconsin Territory, where slavery was banned. Scott sued in federal court for his freedom, and the Supreme Court ruled that Scott had no right to sue in the first place. Only citizens could sue, and slaves could not be citizens. From the Dred Scott court's opinion:

In the opinion of the court, the legislation and histories of the times, and the language used in the Declaration of Independence, show, that neither the class of persons who had been imported as slaves, nor their descendants, whether they had become free or not, were then acknowledged as a part of the people, nor intended to be included in the general words used in that memorable instrument. [...]

[Negroes] had for more than a century before been regarded as beings of an inferior order, and altogether unfit to associate with the white race, either in social or political relations; and so far inferior, that they had no rights which the white man was bound to respect; and that the negro might justly and lawfully be reduced to slavery for his benefit.

Dred Scott was a terrible decision, and is widely known as the worst decision in the Court's history. Worse than Korematsu v. U.S., legalizing Japanese internment, and worse than Plessy v. Ferguson, legalizing Jim Crow laws. It was a major cause of the Civil War. Dred Scott was a symptom of the dysfunction and disunion gripping the nation.

Now, let's fast forward to 1866. The North has won the Civil War, reconstruction of the South is going on, and the Civil Rights Act of 1866, 14 Stat. 27 (1866), has extended citizenship rights to the newly freed slaves. But there are doubts in the Congress whether the Constitution allows blacks and freedmen to be granted political rights. A law passed by Congress could be reversed by a new Dred Scott, or a future Congress could decide to strip the citizenship of the freedmen by a future law.

The solution, of course, was to enshrine those protections into the Constitution itself, which was ultimately ratified by 1868.

This gets us to the relevant text:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.

It means what it says: if you were born in America, you're an American citizen, full stop. This includes children born in the United States to foreigners, United States v. Wong Kim Ark, 169 U.S. 649 (1898), but not the children of foreign diplomats, Slaughter-House Cases, 83 U.S. 36 (1873), nor "Indians not taxed", Elk v. Wilkins, 112 U.S. 94 (1884). (Later, Indians were granted citizenship by the Snyder Act in 1924, rendering that question moot.) This is generally in line with New World citizenship philosophy, but distinct from what generally happens in the Old World, where citizenship is transmitted by blood.