This is a quite old question (more than 3 years now), and chances are OP will never read this. However, I'd like to give my contribution:
The issue here is not if birds incubate or not their eggs (see the other answer). Even if we suppose that no bird incubate its eggs, that is, that all birds lay their eggs and just walk away and that the eggs develop normally at cold temperatures, even so there could be no "marine bird" the way OP asks, like a dolphin. And the answer is in the egg itself.
The amniotic egg
Birds, which are a group of dinosaurs, together with the other reptiles and the mammals show an outstanding evolutionary innovation: the amniotic egg.
The amniotic egg allowed our Amniota ancestor to leave the water and to complete the whole reproductive cycle on land. It was a great evolutionary achievement, which allowed the tetrapod "conquest" of dry land.
An amniotic egg (tortoise)
However, there is a catch: that amniotic egg, which allowed our ancestors to be independent of water for reproduction, cannot be laid on water. Due to several reasons, mainly the (lack of) gas exchange through its shell, the embryo will not develop if the amniotic egg is laid on water.
That's why marine turtles have to leave the water to lay eggs, slowly crawling on the sand in a very exhausting task which will some time later put their very babies in danger. It would be preferable laying the eggs in the water, where they are already... but they can't.
Evolutionary solution: Viviparity
So, how could some mammals take that evolutionary route? Because (most of) mammals don't lay amniotic eggs: they developed viviparity instead. And that viviparity allowed mammals such as whales and dolphins to spend their whole life in the water, never coming to the land.
A very interesting proof of this evolutionary explanation is the case of the ichthysaurs. This extinct group of marine reptiles spent all their reproductive cycle in the water, never returning to the land. How is that possible with an amniotic egg? The answer is that ichthyosaurs didn't lay eggs: just as mammals, they were viviparous. That viviparity allowed them to be truly aquatic.
Artistic representation of an ichthyosaur with its viviparous embryo
This now famous image (Motani et al., 2014) shows a very rare — and tragic, since they died — fossil, a pelvis of a Chaohusaurus mother with three embryos, one of them clearly visible:
So, we can only speculate that, the day some bird develop viviparity (or radical changes in the amniotic egg structure and physiology), then we can have a truly marine bird.
Source: Motani, R., Jiang, D., Tintori, A., Rieppel, O. and Chen, G. (2014). Terrestrial Origin of Viviparity in Mesozoic Marine Reptiles Indicated by Early Triassic Embryonic Fossils. PLoS ONE, 9(2), p.e88640.