The United States of America is today one of the few countries in the world that grants citizenship to everyone born on its soil. Thus, all those born or who were born in the U.S. (with the sole exception of children born to diplomat parents from third countries) are American citizens.
Therefore, if a Greek woman travels to the U.S. with a tourist visa, the visa waiver program, or by any other means and gives birth during her visit, the child born will automatically be an American citizen. This, however, does not mean that the child’s parents acquire any right to stay in the U.S. – the parents of an American citizen gain immigration rights only when the child turns 21 years old, and this under certain conditions.
There are also several practical issues when a foreign woman comes to the U.S. with the intent to give birth. Border officers generally do not look favorably upon foreigners who come to the U.S. specifically to give birth and are likely to deny entry to anyone arriving with such intentions. Finally, giving birth in the U.S. is expensive – in the case a cesarean section is needed, the cost can exceed \$20,000.