No. One needs a valid passport to travel internationally. A birth certificate is no substitute and ads nothing.
A parent who frequently crosses the border by land with a minor should always carry a letter of permission from the other parent. A parent with sole custody of the child they are traveling with should carry a copy of the custody document.
You should obtain a child travel consent letter granting you permission from the absent parent to leave the country with your child (having it notarized is not required, but highly recommended).
The letter of consent, preferably in English and notarized, from the other parent (if the child is traveling with one custodial parent) or signed by both parents (if the child is traveling with a guardian or alone) should say: "I acknowledge that my child is traveling outside the country with the name of the adult ...
Do you need a notarized letter to travel with a child alone? In most cases of a single parent traveling out of the country with their child, the answer is yes you do need a notarized consent to travel letter. You can download your own customizable travel consent letter here.
In the USA, though, unless there's a court order requiring permission from both parents, a child may travel (even to overseas) with either parent/guardian without showing proof of permission from the other parent.
I, ______________________________________________ (name of parent), am the ______ (mother) ______ (father) of __________________________________ , aged ____________ , and do hereby give my consent for (him)(her) to travel with __________________________________________________________________ (name/address of traveling ...
A letter of consent, preferably in English and notarized, from the other parent or signed by both custodial parents. The letter should say: "I acknowledge that my child is traveling outside the country with the name of the adult with my permission."
There is no documentation requirement for domestic travel in the US for children flying with a parent. I've flown dozens of times with no one saying boo.
Typically, minors under the age of 18 do not have to present identification for domestic U.S. travel. Airlines will accept identification from the responsible adult on behalf of the minor(s). For international travel, minors under the age of 18 must present the same travel documents as the adult.