Gounod's Faust was the opera with which every New York "season" began during Edith Wharton's time in the late 19th century. This corresponds to the New York season in which the main characters of The Age of Innocence would coincide for their nightly social entertainment. However, Edith Wharton cleverly uses this fact as a conduit to correlate the themes of The Age of Innocence, to those of Gounod's Faust.
The most salient themes which are shared by both works are the trinity of temptation, followed by passion, and ending up in regret. Newland's temptation to break with the paradigms of his time come with the entrance of the enigmatic and much seductive Ellen into his life. A woman of a modern and more open mentality, Ellen Countess Olenska is shun by those of her social rank for her apparently scandalous divorce. She also breaks with social paradigms; she initiates conversations with men, she leads an independent life, and she is not ashamed of it.
A man like Newland finds in Ellen precisely what his name implies; "a new world" with which he finds himself surprisingly acclimatized. The ideas with which he grows up seem like nothing compared to the new maxims that he discovers through Ellen. Unable, although wishful, to change, Newland chooses the righteous path that detours him from Ellen and plunges him into the social conventions that are expected of him: to marry a prudish woman, to be a good husband, to become a provider, and to make a family. However, like Faust, he realizes that he, nor anybody, is born with the cosmic gift of always making the right choice.
For Newland's choice of perpetuating the social impositions of his time was the wrong choice, no matter how "right" he thought that it would be. Like Faust, Newland's choices are made out of the darkness of ignorance; in reality, Newland does not know anything but what occurs within his immediate surroundings.
All this being said, both Faust and The Age of Innocence present the cycle of temptation, passion, and regret. In Newland, these things come to life after he meets Ellen. In both works, the main characters make the wrong choice, although they seriously considered every variable and determined that the choice would be the correct one. However, in the end, both end up disappointed.
Therefore, we could conclude that Newland certainly sees, in Faust, a story that he can relate to in terms of the want of giving into what is most new and enigmatic to him. However, there is more to the structure of these themes that connects these two literary pieces together, and it is mostly based in the way that the topics of temptation, passion, and regret are used.
The above answer would be very beneficial had the question been about character traits and analysis. The thematic answer may or may not necessarily lead to further questions about character traits. Otherwise, that would have been the initial inquiry which (it is agreed) would deserve lots of additional consideration.
Moreover, the user was clear in that the question was about Gounoud's Faust the opera and not about Goethe's Faust.
Wharton's correspondence with her German governess has just been published.
HerAppleness is correct that the novel refers only to Gounod's opera, but Wharton reportedly knew German and was well versed in the classics 19th-century literature including, in all probability, Goethe's poem.
My insistence on "character traits and analysis" reflects a bias for poetics (i.e., the representation of plot form, of which character traits are essential to the surface causality). Thematics is a perfectly legitimate approach, but would strengthen its interpretation by addressing in its own way the complex form which embodies the idea. The most rigorous model is Eugene Falk's book on thematic structure, which carries an introduction by none other than Bernard Weinberg, a major figure in the Chicago Neo-Aristotelian movement.
I am now tempted by the possibility that Newland Archer and Ellen Olenska are both analogized to Faust, Archer negatively because he cannot choose or act against convention, Olenska positively because she can and does, repeatedly. Each suffers as a result, but in a distinctive and appropriate way.
Very thoughtful thematic answer, but one that invites further questions.
What concrete analogy is possible between Newland Archer's plot form and that of Faust (Goethe's or Gounod's)? Does Archer -- really, can he -- make a morally questionable pact with anyone or anything capable of delivering him from thwarted desire as well as boredom? Is he as provincial and unsophisticated as the reponse suggests? Why, for example, does Wharton imbue him with anthropologically informed skepticism about New York society and his place in it? What of the novels anticlimactic but highly significant finale, which seems to echo "The Love Song of J. Alfred Prufrock" (prestigiously published before Wharton began writing The Age of Innocence) rather than any version of the Faust matter?
Does the Faust analogy really apply to Olenska?