You may think the answer is obvious, the US Constitution states clearly
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
Therefore according to US law only someone born within the USA can be President. This was probably a very sensible precaution in the 1700s. The Republic was recently established and still faced a lot of opposition. There is probably no force on this planet more devious or ruthless than the British nobility and it seemed likely that they would not accept defeat in war lightly. Indeed that they would do something to undermine the new regime was a serious threat and hence taking steps to limit spies and infiltrators was a wise move.
The question however remains why that provision is still in place today. The Framers of the Constitution were not opposed to those who became Citizens after being born elsewhere, otherwise why include that clause "At the time of the adoption of this Constitution"? There is a technical difficulty in that everyone in the US at that time was officially born as the subject of a foreign power and this provision therefore permitted its immediate adoption and execution rather than having to wait 35 years for a person eligible to become President to be born and reach the required age. However I think it was more than that, the Framers were concise and precise with language. If they wanted to they could have limited eligibility to only those born within the geographical area of what was to become the United States but they did not. Anyone who was a Citizen of the US was eligible to stand for President at the time the Constitution was formally adopted regardless of where they were born.
When you think about it, any person meeting that qualification had already proven their loyalty. They had adopted America as their home country and had been involved in a conflict with a foreign power to win independence. It seems only right they should be permitted to stand for the Presidency. Yet everyone who entered the US after that date or became a Citizen must be viewed with a certain caution and suspicion. There was a possibility of foreign agents usurping the political process and it was prudent to prevent this happening. Does such provision really need to be in place today though?
The Queen of England has surrendered a large amount of foreign territory during her reign, much of it peaceably. It is inconceivable that Britain would now seek to gain political control of the US and reverse independence. The UK is a staunch ally of the US during all recent conflicts including Afghanistan and Iraq. However when it comes to size and influence the UK is now very much the junior partner. No one could imagine it taking the US back either militarily or by political interference. Even if in 1787 the UK had the resources to fund a network of spies and infiltrators it no longer does being in the midst of an economic melt down. When the former Chief Secretary to its Treasury wrote there was no money.. he was only partially joking. So the UK is no threat. The US is no longer a new country, it is the only surviving Superpower, unrivalled across the planet, a shining beacon of democracy .. except when it comes to choosing its leader.
If in a couple of years time, I take the Citizenship examination and finally the oath, I will be pledging myself mind and soul to my new country, I will be giving myself wholly to America. However America will not be giving itself to me. I will always be a second class citizen with one key right denied me compared to those who were born here. Now I have no desire to be President but this denial irks me because of its symbolic importance more than its practical effect. It sends a message to the rest of the world that America is still afraid of them. That Americans can never wholly accept and be at peace with someone who was not born there.