The answer is yes and no.
'Spicy' means seasoned by spices. In this case, almost all savory Indian food is 'spicy' as almost all of it is cooked with at least one spice! So the answer is Yes.
However, most people think of 'spicy' and 'chili hot' in the same vein. In this case, to most people's surprise, the answer is No, not all Indian food is spicy!
How Chilies Are Used
Unfortunately, Indian food has garnered a bad name with regards to spice and chili, courtesy some bad cooks who add it to everything! A good Indian cook will only ever add ingredients that enhance each other and the finished dish. Good Indian food is a fine balance of flavors (salty, sweet, sour, bitter, hot) and textures... not too much of anything, chili included. Numerous Indian dishes contain no chili or pepper whatsoever!
While most Indian cooking of savory dishes calls for the use of spices, chilies are not used in all Indian dishes. Some, especially those from South India, are seasoned with pepper too. Each gives its own brand of heat to the dish it is added to. The chilies that are used—dry red chilies, fresh green chilies, red chili powder—each has a different potency and heat. Some dishes rely heavily on chili for their flavor. Some examples are Rasam (pepper water), Vindaloo, Sorpatel) so if you don't like the heat, the only solution is to reduce the amount of chili you use. While this is not ideal and will reduce the authenticity of the dish, better the dish be less authentic and you can eat it than it is 100% authentic and too hot for you. Right?
In other dishes, chili is only used to add the amount of heat which enhances other flavors. Mutter Paneer (peas and cottage cheese) and Rajma (red kidney bean curry) are some examples. In this case, you can omit the chili altogether.
If you have unknowingly added too much chili to a dish, there are still ways to reduce the heat! You could stir in coconut milk and reheat or add potatoes and cook till they are done (potatoes will absorb some of the chilies). You can also eat a hot curry with yogurt to cut some of the heat. This is a trick that's great for kids!
At the end of the day, feel free to experiment with flavors. That is the beauty of Indian cuisine. It is already a melting pot and adapts very well to experimentation!