The situation in the West Indian colonies is an interesting subject, and the key work on the colonies during the war is A. J. O'Shaughnessy, An Empire Divided. It's a thorough and well researched book that talks extensively about the situation in the Caribbean and the lead up the break with Britain.
First, it is important to note that the West Indian colonies did consider rebellion. The islands did dislike some of the trade restrictions and taxes of the the 1760s and 1770s. St. Kitts was the most disruptive, and even had Stamp Act riots to match with the North American colonies. The Leeward islands in general were smaller islands, with less rainfall, and lower overall elevation. The islands were ideal for growing sugar, but the absence of mountainous areas like on Jamaica, made the planters less likely to plant food crops. Almost all the arable land was planted with sugar cane, and the majority of the food stuffs, fish, and timber were imported from North America. The trade between North America and the Leeward solidified commercial and personal ties, making the islands more responsive to the troubles in North America. Further, Jack Greene, a leading historian of the West Indian colonies, and argues that the West Indian colonists had identical ideas about liberty and empire as their compatriots in North America (Greene, Pursuits of Happiness, idem, Peripheries and Center). All this to say that the colonies certainly appreciated what the North American colonies were upset about, and even considered rebellion.
Here are the key differences:
Demography. The white island colonists were outnumbered 10 to 1 by their enslaved peoples in every colony. The only places in North America that had close to this was the Lowcountry of South Carolina, and a few counties in Tidewater Chesapeake. (Morgan, Slave Counterpoint). The West Indian colonists relied on a British military presence to keep their enslaved populations under control. Jamaica had suffered a small scale slave revolt (Tacky's Revolt) in 1760, and relied on British soldiers to help put down the revolt and keep the unrest at a minimum. Edward Long reports of another slave revolt scare in July 1765 (History of Jamaica, 2:465). The colonists did live with a real fear that their slaves would revolt, and would be in especial danger if the colonies evicted British Soldiers. And once the war broke out, Jamaican slaves did revolt in 1776 (O'Shaughnessy, p. 36).
Geographic isolation/danger. The North Americans had the continent or the Appalachian mountains at their back, and the sea at their front. With the eviction of the French soldiers after 1763, and the relative distance to the Spanish colonies in the West, the North American colonists were protected by their natural geography. The West Indian colonies were under no such protection. The Spanish colonies of Santo Domingo, Cuba and posts in South America were a short sail from the British Islands. The French also saw the advantage in trying to take back the Ceded Islands - so called because the French ceded them to Britain in the 1760s (Granada, Dominica, Saint Vincent and the Grenadines). The French had outposts in Saint Domingue and Martinique to raid the British islands. Over the course of the seventeenth and eighteenth centuries, islands across the Caribbean changed hands dozens of times, and the persistent naval presence meant that the West Indian colonists did not want to lose British protection.
Motivations. A few items that were large grievances for the North Americans were either non-consequential or beneficial to the West Indians. First, the Sugar Act of 1764 was a law issued by Parliament directed to protect the interests of the British sugar colonies. North Americans had been trading with the French colonies in sugar and molasses to avoid customs and get lower prices, and the new restrictions of the 1764 act chaffed many of the North American merchants (O'Shaughnessy, p. 67). Second, as mentioned above, the West Indian colonists did not have a natural hostility to standing armies or British soldiers, because the soldiers worked to protect the planters from the enslaved population. (Note - the last major slave revolt in North America was the Stono Revolt in 1739). The Quebec Act (1774) was an act for toleration of the French Catholics that had little impact in the Caribbean, due to the small population of French Catholics in Barbados, Jamaica, and the Leeward Islands. But in North America, and especially in New England, this was a major grievance (Greene, Exclusionary Empire, pp. 168-9).
All in all, the West Indian colonists could understand the grievances of their North American counterparts. However, they found far more advantages to staying within the British Empire, from her soldiers, ships, and trade protections, that they would prefer to live under Imperial restrictions than rebel with North America.