Honda accord error code P0420?

My honda accord 98 LX 4 Cyl 2.3 L have 91000 miles on it. It is showing error code P0420, so i took it to do the engine tune up, and i changed the spark plugs and did engine flush etc, after that the light again came on, and this time mechanic asked me to change the oxygen sensor, but the light again came on, so he again replaced the new sensor with my old sensor and refunded the money for new sensor, but now he is saying something may be wrong with catalytic convertor, and only if it genuine honda, only then the code will go, otherwise the code will be there, so please give me some advice about it, and he also told me if your car is running fine i dont have to worry about it, but i want my car to be perfect, i dont like that engine light on, all the time, please give me a advice, will bad catalytic converter hurt my car.....