This whole Gaudiya tradition is a very late, highly polemical and sectarian thread of Vaishnavism that has very little relation to Vedic thought.
Srila Prabhupada himself stated that he did not "scrutinizingly study" the entire Vedic corpus, but instead focused on the Bhagavad Gita and Srimad Bhagavatam (Sometimes called Bhagavata Purana) which is Puranic literature. In ISKCON to this day it is these two scriptures that are emphasized, along with the volumes of teachings and biography of the Saint Chaitanya, Chaitanya Charitamrita(Chaitanya is regarded by Gauudiya Vaishnavas as an incarnation of Krishna, but not by most other Hindus). So ISKCON is Vedic in the sense that it bases its philosophy around two important Vedic Scriptures.
So we could say that ISKCON is selectively Vedic- simply it has its points of emphasis, as do many traditions within Hinduism because the Vedas are so impossible vast. Just as Pure Land focuses on one genre of Sutras, or Tiantai, or Madhyamika (at least according to the Gelug interpretation) for that matter, which accepts the second turning of the Wheel and its teachings as definitive and the others as provisional. In fact, I would argue that especially in Mahayana Buddhism where there is an almost impossibly vast corpus of literature, there are approaches amongst the various schools of Buddhism to emphasize the important points or present what they see as the essence of the sutras in commentarial literature. In Chinese Buddhism and Japanese Buddhism you see schools that advocate exhaustive study of one Sutra as the ultimate sutra- especially in the Huayen, Pure Land, Tian Tai and various Nichiren schools.
My main point of contention with Bhaktivedanta Swami (Prabhupada) is that he seems to present a vision of the Vedas that is all embracing, and almost wishy-washy here: http://www.srilaprabhupad.com/dloads/20 ... ations.htm
Veda means knowledge. Vetti veda vido jnane. Vid-dhatu is called veda, vetti. Jnane when there is question of knowledge, these three forms are used: vetti, veda, vido, jnane. Vinte vid vicarane vidyate vid saptayam labhe vindati vindate. (?) This is the vid-dhatu description. So vid-dhatu means to know. So ultimate knowledge is to know God.
So God is purest. Param brahma param dhama pavitram paramam bhavan [Bg. 10.12]. How one can approach God if he leads a sinful life? That is our propagation. You give up this sinful life. Then you'll be able to understand God. You follow Christianity or Mohammedanism or Buddhism. It doesn't matter. You give up this sinful life.
Then in many other passages in his writing Prabhupada insists that the only way to truly approach god is through Srimad Bhagavatam, as only it reveals the supreme aspect, or personality of god-head: http://www.prabhupadavani.ca/main/Bhagavatam/004.html
So Krsna appeared for reestablishing the real principles of dharma, or religion. So He did not come or did not appear for establishing the so-called religious system, Hindu religion and Muslim religion or Christian religion or this religion, that... Not that type. Real religion. Therefore He says, sarva-dharman parityajya: [Bg. 18.66] "Give up all these. Simply surrender unto Me." So any religious system which is teaching to divert the attention of the follower to so many things, that is cheating religion.
Prabhupada's books are interesting but his sectarianism is unfortunate, with acidic criticism of "Mayavadis" (impersonalists, who are actually proponents of Advaita Vedanta and other fascinating, sophisticated strands of Hinduism) coming page after page. This means that unfortunately many well-translated passages and interesting commentary may be disregarded in the field of Indo-religious studies because such rhetoric would lead one to think there is a strong bias.
The amount of commentary and translation produced by Prabhupada and his disciples is amazing, and I respect the huge amount of discipline required to really be a follower of his teachings. However, the "Mayavadi" philosophers also have a great deal of interesting ideas, as do proponents of philosophies like Saiva Siddhanta. So if one is interested in Gaudiya Vaishnavism as a path Prabhupada's books are all that is required, but if one wants to understand in a broad way Hinduism or Sanatana-dharma, it is necessary to read other texts and commentaries.