You cannot run a #10 (10-gauge) wire from a 30A breaker to 15A/20A receptacle.
You also cannot run that wire to a junction box for a 15A/20A appliance hard-wire, unless the appliance allows it and it is mentioned in the manual (rarely).
Whether you need to wire to an outlet with or without receptacle depends on the microwave. Does its cord have a plug?
You can run a #10 to a sub panel, then from a 15A or 20A breaker by #14 or #12 wire to an outlet box with a receptacle for a plug-in, or just a junction for the hard-wire.
You can replace the breaker with a 15A (or 20A), use the existing #10 to where it terminates, and use #14 (or #12) wire to continue to a 15A receptacle.
The sub-panel will require dedicated wall space free of obstructions, and it's more work as it requires a wall cutout. All this depends on your circumstances in the kitchen or a nearby space. For instance, is there a garage behind the kitchen?
If your microwave has a plug, you can also run a #10 to a 4-prong stove receptacle (NEMA 14-30), then a "gas range adapter plug" with built-in 15A protection to provide a 15A 120V receptacle for the microwave plug.
You can wire a 15A appliance via #14 "tap wire" from a #10 protected by a 30A breaker if the tap length is 10ft or less, and the appliance has built-in overcurrent protection. The manual would state so, and it wouldn't be with a plug.
In general
- You can wire a 15A receptacle to a 15A or 20A circuit, and you can over-gauge the wire, e.g. use an existing #10.
- You cannot wire a 15A receptacle to a 30A circuit, regardless the wire gauge.
The appliance also has a cord and internal circuitry that require protection at the 15A or 20A level by a breaker. Any allowable exceptions would have to be listed in the manual.
If all this sounds too complicated, you may be better served with help from an electrician on-site or a DIY-friend with electrical experience.