Yes, they do damage habitats, namely from bark stripping, and disease. This is why the UK class them as a pest. However it is not limited to grey squirrels as red squirrels also strip bark, but they have much smaller numbers proportionally so their damage is proportional. Lack of grey squirrels therefore does not wholly prevent this damage.
In addition to displacing this native species they frequently cause damage to woodlands by stripping bark from the main stem and branches of trees
This has an effect commercially and natively. This can lead to a loss in food for red squirrels, birds, dormice etc.
Such damage acts as a major disincentive to the planting of broadleaved and coniferous trees for timber as it reduces the value of the final crop. Increasingly wider impacts are being recognised as potentially of major significance to woodland conservation, biodiversity and sustainability. In time damage may lead to a loss of particularly vulnerable species (e.g. beech) within the mature canopy of woodlands and this may be accompanied by loss of associated fungal and invertebrate fauna and their predators. In addition, there may be indirect competition, e.g. for food, between grey squirrels and native fauna such as the red squirrel and common dormouse; or predation by grey squirrels, e.g. on woodland bird populations. Grey squirrels also carry squirrelpox virus, an infection fatal to red squirrels.
The UK has been controlling the population of grey squirrels for many years, and we have some grey free areas, such as Brownsea Island.
