Yes, you can certainly execute your tests on mobile devices using selenium webdriver.
You can use Appium(http://appium.io/) server to do that.
You can test web applications as well as native moible applications for Android and iOS both.
Setting up desired capabilities:
DesiredCapabilities capabilities = new DesiredCapabilities();
capabilities.setCapability(MobileCapabilityType.PLATFORM_NAME, "iOS");
capabilities.setCapability(MobileCapabilityType.PLATFORM_VERSION, "7.1");
capabilities.setCapability(MobileCapabilityType.BROWSER_NAME, "Safari");
capabilities.setCapability(MobileCapabilityType.DEVICE_NAME, "iPhone Simulator");
Sample of the web application test:
DesiredCapabilities desiredCapabilities = new DesiredCapabilities();
desiredCapabilities.setCapability(MobileCapabilityType.BROWSER_NAME, "Safari");
URL url = new URL("http://127.0.0.1:4723/wd/hub");
AppiumDriver driver = new AppiumDriver(url, desiredCapabilities);
// Navigate to the page and interact with the elements on the guinea-pig page using id.
WebElement div = driver.findElement(By.id("i_am_an_id"));
Assert.assertEquals("I am a div", div.getText()); //check the text retrieved matches expected value
driver.findElement(By.id("comments")).sendKeys("My comment"); //populate the comments field by id.
//close the app.
driver.quit();
Sample of the native application test:
Sample APK:
Eg.
wait(for_text("Accessibility")).click();
wait(for_text_exact("Accessibility Node Provider"))
Hope that helps