What does the poem have to say about different ways of looking?(One may want to consider the relevance of the concept of the "male gaze" to this poem.)
By asking this question, you bring up a fresh way of analyzing this poem. Let’s look at the direct references first: The Duke is giving the ambassador a sort of tour of his mansion (you might ask why); he stops on a landing and draws a curtain (“since none puts by the curtain I have drawn for you but I”), revealing a portrait of his “last” duchess; after making some revealing remarks there, he then goes further down the stairs, and makes a point of Claus of Innsbruck’s sculpture of Neptune and a seahorse.
These are the primary “looking” moments. Going back to the portrait, we “see” that the Duchess was beautiful, a little shy, wearing a shawl. Then we “see” the Duke’s “male gaze”; he was jealous of every little attention paid to her, by the gardener, by the portrait artist (Fra Pandolph, a friar no less!); but most importantly, we see the pride of the duke (“As though she ranked my gift of a nine hundred old name with anybody’s gift!”).
The best “seeing” is our (and, we assume, the ambassador’s) seeing through the duke’s outer haughty sophistication to his small, cruel, avaricious (“no just pretence of mine for dowry” ego. Every detail Browning provides builds our “seeing through”: “taming a seahorse” is a very subtle clue to what he did to the duchess (“then all smiles stopped together”) So, the “male gaze” idea hold weight, especially if we include the ambassador, Claus of Innsbruck, Fra Pandolph, the gardener, and ourselves into the “seeing.” "Seeing" can mean "looking at things carefully," or "psychological point of view" or "understanding," or "gender-based bias," among others.