OK, here it is from the US Governements mouth so you can all stop speculating on citizenships.
In order for Prince Harry to obtain US Citizenship, he would have to renounce his titles.
"C. Renunciation of Title or Order of Nobility
Any applicant who has any titles of heredity or positions of nobility in any foreign state must renounce the title or the position. The applicant must expressly renounce the title in a public ceremony and USCIS must record the renunciation as part of the proceedings. [5] Failure to renounce the title of position shows a lack of attachment to the Constitution.
In order to renounce a title or position, the applicant must add one of the following phrases to the Oath of Allegiance:
I further renounce the title of (give title or titles) which I have heretofore held; or
I further renounce the order of nobility (give the order of nobility) to which I have heretofore belonged. [6]
An applicant whose country of former nationality or origin abolished the title by law, or who no longer possesses a title, is not required to drop that portion of his or her name that originally designated such title as a part of his or her naturalization. [7]"
www.uscis.gov/policy-manual/volume-12-part-j-chapter-2
In order for Prince Harry to even start to become a US citizen, there are residency requirements.
"General Eligibility Requirements
To be eligible for naturalization under section 319(a) of the INA, you must:
Be at least 18 when you submit Form N-400, Application for Naturalization;
Be a lawfully admitted permanent resident of the United States for at least three years immediately before the date you file Form N-400;
Have been living in marital union with your U.S. citizen spouse during the three years immediately before the date you file your application and while we adjudicate your application;
Have lived for at least three months in a state or USCIS district having jurisdiction over your place of residence;
Have continuous residence in the United States as a lawful permanent resident for at least three years immediately before the date you file your application;
Reside continuously within the United States from the date you filed your application until the date you naturalize;
Be physically present in the United States for at least 18 months out of the three years immediately before the date you file your application;
Be able to read, write and speak English and have knowledge and understanding of the fundamentals of the history, and of the principles and form of government, of the United States (also known as civics); and
Be a person of good moral character, attached to the principles of the Constitution of the United States, and well-disposed to the good order and happiness of the United States for at least three years immediately before the date you file Form N-400 and until you take the Oath of Allegiance.
For more information, see the USCIS Policy Manual, Volume 12, Citizenship and Naturalization.
For information relating to spouses of military members, see our Citizenship for Military Family Members page.
For information about becoming a lawful permanent resident (LPR) or petitioning for family members, please visit our Green Card or Family pages."
Harry can apply for a Green Card due to his marriage to a US citizen. Once approved, he will become a Long Term Resident.
www.uscis.gov/green-card/green-card-eligibility-categories
Once he becomes a permanent resident, he has the following rights and responsibilities (he does not have to renounce his titles)
Your Rights as a Permanent Resident
As a permanent resident (Green Card holder), you have the right to:
Live permanently in the United States provided you do not commit any actions that would make you removable under immigration law
Work in the United States at any legal work of your qualification and choosing. (Please note that some jobs will be limited to U.S. citizens for security reasons)
Be protected by all laws of the United States, your state of residence and local jurisdictions
Your Responsibilities as a Permanent Resident
As a permanent resident, you are:
Required to obey all laws of the United States and localities;
Required to file your income tax returns and report your income to the U.S. Internal Revenue Service and state taxing authorities;
Expected to support the democratic form of government (“support” does not include voting. Permanent residents cannot vote in federal, state, or local elections.); and
Required to register with the Selective Service, if you are a male age 18 through 25.
For more information, see the Maintaining Permanent Residence and International Travel as a Permanent Resident pages.
There are many, many permanent residents in the US who never naturalize. So it is not unusual.