OP: Are there any arguments Protestants use against scriptural authority that are common to both Catholic and Orthodox Churches?
There are basically three phases of pre-1054, post-1054, and Reformation.
Keep in mind that prior to 1054, the church wasn't divided into Eastern Orthodox and Roman Catholic. So, whatever argument might arise about Catholic scriptural authority would also apply to Orthodox scriptural authority.
Post 1054, there are numerous differences between the two denominations, but they both can't be right on the same issue, so how does one know either is right on any issue?
So, what are some arguments?
This is from a letter from Firmilian to Cyprian regarding Pope Stephen's agreement that baptism by heretics is identical to baptism in the Church.
- But that they who are at Rome [Stephen] do not observe those things in all cases which are handed down from the beginning, and vainly pretend the authority of the apostles;2925 any one may know also from the fact, that concerning the celebration of Easter, and concerning many other sacraments of divine matters, he may see that there are some diversities among them, and that all things are not observed among them alike,
Anyone may know that Rome (Catholic or Orthodox) does not follow apostolic teaching and other sacraments, while vainly pretending to have apostolic authority.
After 1,756 years, the Catholic Church finally came around, sort of. We're still not sure how it views other baptisms by heretics.
The Vatican ruling, he said “indicates that the Catholic Church also recognizes that LDS baptism is substantially different from Roman Catholic baptism.’’
At the 1054 split, both sides claimed apostolic authority, but obviously the apostles didn't teach contradictions.
What is interesting is that each side (Catholic and Orthodox) will claim apostolic succession, but again, each will teach something contradictory to each other. This means that over the years, they redefined what "apostolic succession" means. It changed from "teach the same to faithful men" to "trace your so-called lineage and teach whatever you want".
And the things that thou hast heard of me among many witnesses, the same commit thou to faithful men, who shall be able to teach others also. 2 Tim 2:2
And of course there is the Reformation with all of its unanswered arguments against what Rome was teaching.
So, anyone may know, not just Protestants, of vain claims to authority and failed apostolic succession to teach the same.