Disable / Enable Chrome Extension Via Browser Action / Icon

Such API is not provided. But two possible workarounds exist:

I. You can use the "disabled" flag variable and update it from your background page.

Background page:

function disableExtension(disabled)
{
    chrome.windows.getAll({populate : true}, function (window_list)
    {
        for (var i = 0; i < window_list.length; ++i)
        {
            var window = window_list[i];
            for (var j = 0; j < window.tabs.length; ++j)
            {
                var tab = window.tabs[j];
                if (checkContentScriptExists(tab))
                {
                    chrome.tabs.executeScript(tab.id, {code : "disabled = " + disabled + ";"}, allTabs: true) 
                }
            }
        }
        // No matching url found. Open it in the new tab
        chrome.tabs.create({ url : url, selected: true });
    });
}

And content script should check the condition before the run

if (!disabled) doSomething();

II. A controversial approach to save disable variable within background age content

Background page:

function disableExtension(disabled)
{
    global.disabled = disabled;
}
chrome.runtime.onMessage.addListener(function(request, sender, sendResponse) {
    if (request.msg == "getDisabled") {
        sendResponse({disabled: global.disabled});
        return true;
    }
});

and the content script should query currently disabled status before execution

chrome.runtime.sendMessage({msg: "getDisabled"}, function(response) {
   if (!response.disabled) doSomething();
});