Can we do Pulse width measurement without input capture method?
Is any any method?
How accurate this method will be?
Busy loop sampling will add delay?
On one hand, it will require the main loop to enter a busy loop delay.
But when it comes to timing precision, it can manage much better than 10us resolution because almost any modern processor will manage a compare/repeat loop at much higher iteration speeds than 10us. A busy loop is likely to manage 0.1us or better resolution with most ARM chips - as long as there aren't interrupts that burns lots of us.
Does the RX610 have the Debug/Trace option implemented? Could always time stamp with DWT_CYCCNT
Other than that you'd want to look for a 32-bit (or 16-bit) counter register that you can read and make measurements against.