Green ideology is rooted in the idea of absolute limits to human progress and a related idea that attempts to exceed these (often relatively arbitrary) limits are somehow sinful. I consider that more right-wing than left-wing. It says "know your place and stay there" not "try to make something better."
'Conservativism' is different than 'right wing,' and 'liberalism' is different than 'left wing.'
The standard polisci definition is that (small 'c') 'conservatives' seek slow organic change in society while (small 'l') 'liberals' seek deliberate and intentional change. In addiction we have 'radicals,' who seek total change and an overthrow of the current system, and 'reactionaries' who either want no change, or to reverse the flow. ('Radical conservative,' is an oxymoron') Big 'C' Conservatives and big 'L' Liberals are different still, (and both are different than 'Progressives.')
Greenpeace is radical, left-wing movement. They are such in the same way that both the French Revolution and Bolshevism were radical left-wing movements.
Western notions of "human progress" during the last few centuries or so happen to correlate well with resource depletion. Green ideology presents a challenge to these notions of progress, and tantalizes us with the possibility of alternative notions of progress (e.g., sustainable development, permaculture, green technology, etc.)
No, it like or not it belongs on the 'left.'
The difference between 'left wing' and 'right wing' ideologies isn't the presence of a religious nature, secular or otherwise, nor does it have anything to do with any specific policy. The difference is this: 'right wing' ideologies take for their inspiration an idealized past, 'left wing' ideologies take for their inspiration an imagined future.