The prince is quite aware of his skin colour, yes, but is hardly apologetic about it. He appears proud and confident, even arrogant. A careful scrutiny of his opening statement reveals this clearly.

Mislike me not for my complexion,
The shadow'd livery of the burnish'd sun,
To whom I am a neighbour and near bred.
Bring me the fairest creature northward born,
Where Phoebus' fire scarce thaws the icicles,
And let us make incision for your love,
To prove whose blood is reddest, his or mine.
I tell thee, lady, this aspect of mine
Hath fear'd the valiant: by my love I swear
The best-regarded virgins of our clime
Have loved it too: I would not change this hue,
Except to steal your thoughts, my gentle queen.

He asks Portia not to dislike him because of his skin colour, which he describes as a dark uniform brought about by a shining sun of which he is a neighbour. He challenges Portia to provide the most handsome man from the north, where the moon's heat scarcely thaws icicles so that they can cut one another to prove whose blood is the reddest. He implies either that there will be no distinction or that his will be reddest. He further asserts that his skin colour has driven fear into courageous men. He also swears by his love that the most admired virgins of his climate (country) have also loved his skin tone. He exclaims that he would not change the shade of his skin except to determine what Portia felt. It is clearly important for him to know what she thinks of him.

Portia replies by saying:

In terms of choice I am not solely led
By nice direction of a maiden's eyes;
Besides, the lottery of my destiny
Bars me the right of voluntary choosing:
But if my father had not scanted me
And hedged me by his wit, to yield myself
His wife who wins me by that means I told you,
Yourself, renowned prince, then stood as fair
As any comer I have look'd on yet
For my affection.

Portia assures him that she is not wholly directed by good looks and, besides, she is barred from choosing a husband and is guided by chance in terms of her father's will. She furthermore states that if her father had not limited her in choosing a husband and that she has to give herself to one by the means determined in his will, he, the famous prince, would have stood just as much a chance as any man that she had looked upon, to win her affection. Portia here displays an honourable courtesy to the prince and clearly will not offend him. 

The prince thanks Portia for her generous comment and bravely states how he would fight the bravest warriors and attempt the most dangerous deeds to win her heart. He expresses disappointment, however, that winning her hand has been reduced to a mere game of chance in which the weaker could reap all the benefits. He asks to immediately make his choice. Portia informs him about the applicable conditions when choosing a casket and states that he can take the risk after dinner.

The prince eventually ends up choosing the golden casket which is the wrong choice and then, although disappointed, graciously departs, stating:

Portia, adieu. I have too grieved a heart
To take a tedious leave: thus losers part.