Is there any occasion in Shakespeare's play Othello where Othello reflects on his own race, possibly relating to his insecurity?

In Act 3, Scene 3, Othello does refer to his race. He uses the reference in a monologue after the devious Iago has manipulated him into believing that Desdemona was involved in an inappropriate liaison with his lieutenant, Cassio. Iago has, throughout their conversation, made suggestions by using innuendo, intimations and equivocation to encourage the general's jealousy and then, ironically, suggesting that he should 'beware of jealousy.'

Othello is overcome by Iago's scheming and deceit and seeks justification for Desdemona being involved with Cassio. He is quite honest about himself and exposes his insecurities.

He mentions: 

Haply, for I am black
And have not those soft parts of conversation
That chamberers have, or for I am declined
Into the vale of years,--yet that's not much--
She's gone.

The implication here is that it is a matter of chance that he is black and thus lacks the ability to speak softly, gently or sweetly as fashionable or intriguing men do. He believes that since he has not constantly whispered sweet nothings in Desdemona's ear, she might have turned from him. Othello's admission here links to his earlier contention in Act 1 when he also mentioned his inability to use beautiful or florid language, where he stated that he 'is rude in speech.'

Later, in Act 3, Scene 3, when Othello is fully convinced about Cassio and Desdemona's adulterous affair, he again refers to his race. In this passionate statement, he swears to take revenge on them for cuckolding and humiliating him. Iago has convinced him that Cassio, through his behavior, has definitely proven that he is having an affair with Desdemona. He also tells the highly upset general that he had seen Cassio wipe his beard with the napkin Othello had given Desdemona as a gift. Othello passionately cries out:

O, that the slave had forty thousand lives!
One is too poor, too weak for my revenge.
Now do I see 'tis true. Look here, Iago;
All my fond love thus do I blow to heaven.
'Tis gone.
Arise, black vengeance, from thy hollow cell!

The general then kneels and makes a vow to take revenge. Iago kneels with him and promises allegiance. Othello then, out of gratitude for Iago's 'exceeding honesty,' awards him the title of lieutenant, which is the greatest irony of all.