Can LED Driver chips drive the LED with continuous ON DIMMING?
Please look at page 8 of the datasheet under "Dimming Control". You can do either PWM or analog dimming. For analog, you vary the EN/DIM pin between 0.7V (0%) and 1.4V (100%). For PWM, you use a 1HKz signal with on > 1.4V and the duty cycle controls the brightness.

You can do it either way.

And to answer your question, putting 5V into EN should give you 1A to the LEDs continuously.