On the contrary, in Medillin's petition, the Government of Mexico raises the issue of diplomatic protection of its citizens and has asked for an annulment of the conviction and sentence for Medillin. That certainly raises jurisdictional questions and materially affects the right of the State of Texas to prosecute Medillin for his crimes. See this quote from the brief filed on Medillin's behalf in the Supreme Court case:
Seeking relief on its own behalf and, in the exercise of its right of diplomatic protection, of its nationals, Mexico claimed that the United States had violated Article 36 in each of those cases and requested, among other relief, the annulment of the convictions and sentences of the 54 Mexican nationals
Here is Article 36 of the VIENNA CONVENTION ON CONSULAR RELATIONS
Article 36
COMMUNICATION AND CONTACT WITH NATIONALS OF THE SENDING STATE
1. With a view to facilitating the exercise of consular functions relating to nationals of the sending State:
(a) consular officers shall be free to communicate with nationals of the sending State and to have access to them. Nationals of the sending State shall have the same freedom with respect to communication with and access to consular officers of the sending State;
(b) if he so requests, the competent authorities of the receiving State shall, without delay, inform the consular post of the sending State if, within its consular district, a national of that State is arrested or committed to prison or to custody pending trial or is detained in any other manner. Any communication addressed to the consular post by the person arrested, in prison, custody or detention shall also be forwarded by the said authorities without delay. The said authorities shall inform the person concerned without delay of his rights under this sub-paragraph;
(c) consular officers shall have the right to visit a national of the sending State who is in prison, custody or detention, to converse and correspond with him and to arrange for his legal representation. They shall also have the right to visit any national of the sending State who is in prison, custody or detention in their district in pursuance of a judgment. Nevertheless, consular officers shall refrain from taking action on behalf of a national who is in prison, custody or detention if he expressly opposes such action.
2. The rights referred to in paragraph 1 of this Article shall be exercised in conformity with the laws and regulations of the receiving State, subject to the proviso, however, that the said laws and regulations must enable full effect to be given to the purposes for which the rights accorded unde this Article are intended.
Please note Section 2 above - why? Because it confims the receiving State's jurisdiction. And note that the last sentence of 36 1 (b) seems to be at issue in the Medellin case, and doesn't speak at all towards jurisdiction.
Further, please note Article 41 para 3, which affirms the receiving State's jurisdiction over consular officers, by stating that:
3. If criminal proceedings are instituted against a consular officer, he must appear before the competent authorities.
In fact, the only time a consular officer is afforded immunity is during the course of his or her duties, as specified in Article 43:
Article 43
IMMUNITY FROM JURISDICTION
1. Consular officers and consular employees shall not be amenable to the jurisdiction of the judicial or administrative authorities of the receiving State in respect of acts performed in the exercise of consular functions.
2. The provisions of paragraph 1 of this Article shall not, however, apply in respect of a civil action either:
(a) arising out of a contract concluded by a consular officer or a consular employee in which he did not contract expressly or impliedly as an agent of the sending State; or
(b) by a third party for damage arising from an accident in the receiving State caused by a vehicle, vessel or aircraft.
Seeing that Mr. Medellin was not a consular officer, and certainly not acting as one when he committed his crimes, the argument that his case is about jurisdiction has nothing to support it.
Plainly put, the United States has jurisdiction over persons on it's territory. And as things stand right now, that jurisdiction confers American citizenship to those born under it, per the 14th Amendment.