On what grounds was Arius refuted at the First Council of Nicaea? What proof does anyone have that Christ was the same and identical as the one He called "Father" even when Christ himself said in John 14:10...


  Don't you believe that I am in the Father, and that the Father is in
  me? The words I say to you I do not speak on my own authority. Rather,
  it is the Father, living in me, who is doing his work. (NIV)


Yes, Jesus was the Son of God although He never Himself said as much; He said He was "the Son of Man" only (for, I believe, He was truly humble in nature). But how can men know exactly what Christ was without experiencing a gnosis of some kind? The very fact that they proclaimed to have known what exactly Christ was precludes any so-called "evidence" they put forth.

On the other hand, Constantine was a politician/stateman and a duplicitous pagan most of his life.  How could he and his associates possibly know what exactly Christ was and make an absolute statement that He was identical to the one He called "Father" and call the Arian/Arius view a heresy?
The records of the 1st Ecumenical Council do not, I believe, include the discourse that preceded the condemnation against Arius and his followers.  

We do know, however, that Athanasius of Alexandria was key in the condemnation.  He had written extensively against Arianism in his Four Discourses Against the Arians.  We can assume, I think, that many or all of the arguments from this document were brought to bear at the Council and/or that most of the Council members were familiar with them, even though the document itself seems to have been written after the council. I do not think it would be misstating things to say that the 1st Ecumenical Council was essentially an endorsement by the whole Church of a previous deposition of Arius by the bishop Alexander within the See of Alexandria.

I provided a link to Athanasius' document above, which you see is quite comprehensive and detailed, with several sections devoted to exegesis of the New Testament Scriptures, contrasting what was understood to be the Orthodox interpretation with that of the Arians.  An outline of the arguments against Arius is:


The Sonship of Christ is eternal 
The Sonship of Christ is not like earthly Sonship
The personality of the Son is co-eternal with the Father


As you will see in the discourses, Athanasius was quite thorough.  On each point, he considered the Arian's argument, and then considered and refuted the Arian's interpretation of each Scriptural "proof text" offered.