Your Honda's A/C fan only works on high is almost certainly caused by a failed blower motor resistor. The resistor controls lower fan speeds by reducing the voltage, and when it burns out, the fan defaults to high speed, bypassing the resistor entirely.
Note
Important Considerations for Honda A/C Repair
The manufacturer may cover some Honda A/C issues at no cost. Hoover Street Auto Repair will always check your vehicle for active A/C system recalls or extended warranties. If we find problems covered by your warranty, we will advise you to return to the dealer.