According to US Laws, a minor who is born abroad with one parent who is a US citizenship can automatically become a citizen. Is that minor eligible to run for the office of President of United States?

First, if you are asking this question in regard to Barack Obama (who some claim was born in Kenya), the answer is no. In order to be President, you must be a natural born citizen. The law at the time Barack was born was clarified shortly before his birth. The case was Montana v. Kennedy, 366 U.S. 308 (1961). It provided:

A person born in 1906, whose mother was a native-born citizen of the United States and whose father was a foreign citizen, who was born overseas and then moved to the United States, was not a citizen of the United States by birth.

Of course, Barack was actually born in America, so this has no bearing on his eligibility.

Modernly, the Constitution's Natural Born Citizen requirement has been interpreted in a bicameral way. That is: you are either a natural born citizen or a naturalized citizen. Naturalization is the process of applying for citizenship, taking a test, getting a background check, etc, etc.

If you do not have to go through the naturalization process (which foreign born children to an American citizen do not), then under modern law you are a naturally born citizen.

In short, natural born in the Constitution does not mean that you were born in America. Rather, it means you are an American citizen by virtue of being born (i.e cannot be denied to you), and not by virtue of earning citizenship.

No, the US Constitution expressly provides that only 'natural-born' citizens can become president. This is the reason why the current Governor of California can never be president.

This was also in the public spotlight recently when newspapers examined John McCain's eligibility for the presidency. McCain was born in the Panama Canal Zone, after which an Act of Congress retrospectively declared him and other children born in the Canal Zone 'natural-born' citizens.

See, for example the New York Times artice: http://www.nytimes.com/2008/02/28/us/politics/28mccain.html

Adding a bit to the question:

The form is called N-600 "Certificate of Citizenship"

Instructions for the form:

http://www.uscis.gov/files/form/N-600instr.pdf

Application:

www.uscis.gov/files/form/N-600.pdf