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Can we do Pulse width measurement without input capture method?
Is any any method?
How accurate this method will be?
10us resolution isn't the same as being able to detect a 10us interval. You would normally want much higher resolution if the pulse is around 10us long.
Anyway - 10us means 100kHz. So it's possible to use continuous SPI receive at 100kHz to get 10us resolution. In some situations, it's also possible to have the pulse be detected as a start bit by an UART.
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.