Recently a question was asked about the reasons why many Hindus are opposed to the Aryan Migration Theory (AMT)? The question was asked in a manner that does not fit with the requirements of this site. So I have worded the question in a manner that best suits this site. I feel the question is important and it is important to get a scholarly answer that explains the reasons for the opposition.
There is no spiritual principle that is behind the opposition to AMT by many Hindus. The reasons for the opposition is spelled out very lucidly in this site: http://talageri.blogspot.in/2018/04/what-is-value-of-new-genomic-evidence.html?m=1
Sri Srikant Talageri is the principal Indian scholar behind the Out of India Theory (OIT).
The main point is that genetic data can not tell us definitively about language migration. Moreover linguistic theories are in direct conflict with archeology. Linguistic theories say that there was definitely migration into India by Indo-Aryan speakers while archeologists like Kennedy are equally sure that there was no migration by people with different body types in the time frame of the AMT or AIT theory. Moreover textual data from the Rig Veda and archeological data from Mittanis in West Asia are in conflict with the linguistic data. At least this is what I understood.
I am giving more explanation to further clarify the situation. It is not the migration of Indo-European speakers that is the issue. The problem is the insistence of the linguists that the migration occured around 1500 BCE. This creates problems with the Rig Veda since 1500 BCE is also the timeline of the Rig Veda. This raises the question as to why Rig Veda does not have any memory of inward migration into India. Another problem is with respect to Saraswati river. Rig Veda talks of the Saraswati river flowing from the mountains to the ocean. The Saraswati river stopped flowing to the ocean around 1900 BCE. If the Rig Veda was composed by the Indo-Aryan speakers around 1500 BCE the why should the Rig Veda talk of that river flowing to the ocean. There are also other issues that complicate understanding of India's prehistory.
This article http://archaeologyonline.net/artifacts/genetics-aryan-debate gives a good summary of the Aryan invasion debate.
No central Asian genes have been found in Rakhighari skeleton. https://economictimes.indiatimes.com/news/politics-and-nation/harappan-site-of-rakhigarhi-dna-study-finds-no-central-asian-trace-junks-aryan-invasion-theory/articleshow/64565413.cms
Evidence for chariots found in Bronze age dig site. http://www.newindianexpress.com/nation/2018/jun/05/uttar-pradesh-asi-unearths-first-ever-physical-evidence-of-chariots-in-bronze-age-1824138.html
Rakhigarhi skeleton DNA demolishes Aryan invasion/migration theory https://timesofindia.indiatimes.com/india/dna-analysis-of-rakhigarhi-remains-challenges-aryan-invasion-theory/articleshow/71018198.cms?utm_source=email&utm_medium=social&utm_campaign=TOIMobile
Rigveda 9.63.5 says
इन्द्रं वर्धन्तो अप्तुरः कृण्वन्तो विश्वं आर्यं अपघ्नन्तो अराव्णः
Magnifying Indra, quick, making everything noble, weakening off the malignities
Someone else like Aryasamajist may translate it in a different way. It was the translation of my friend, a vedic scholar.
Anyways, the word "Arya" was used in Rig Veda indicating the quality "Noble", but not any race or sect or something else.
In Ramayana also the word Arya was used to indicate courageous/Noble, etc, but not in the sense of race.
Seeing Kumbhakarna coming with his colossal body and emitting a tremendous roar, the monkeys scare away. Angada reassures the monkeys, who then return to the battle field to resume fight.
कुलेषु जाताः सर्वे स्म विस्तीर्णेषु महत्सु च || क्व गच्छत भयत्रस्ताः प्राकृता हरयो यथा | अनार्याः खलु यद्भीतास्त्यक्त्वा वीर्यं प्रधावत || (Yuddha kanda 66 Sarga 21-22 Slokas)
"All of us are born in distinguished races; which are well developed. Where to you go frightened, like ordinary monkeys? As you are running away with fear, leaving all your valour, you are indeed unworthy of honour."
Here, the sage Valmiki used the words अनार्याः खलु (anaaryaah khalu) = you are indeed not worthy of honour. The "Arya" was used here to indicate a person, who is Valiant and cares for his honour.
Coming to the question - Is there any Hindu spiritual principle that causes many Hindus to oppose the Aryan Migration Theory? -,
1) I would like to state that the word "Arya" has been used in different connotation, as explained above, than what Max Müller and others understood or projected to be so.
2) Aryaman, the third Aditya, the one who holds the power of nobility and refinement (Arya), is one of the early Vedic Hindu deities.
Hence, being a Hindu, I cannot accept the different theories on the word Aryan.