In a word, no. You need to provide an Arduino Micro with either > 6.5V unregulated, or a regulated 5V supply.

You could use a buck/boost DC/DC converter to boost your 3.7V battery output to 5V and feed that directly into the 5V line on the Arduino (or better, feed it into the USB connector, since that will use the input switching on the Arduino.

This one, for example:

https://www.banggood.com/DC-DC-3V-15V-to-5V-Fixed-Output-Automatic-Buck-Boost-Step-Up-Step-Down-Power-Supply-Module-p-1355829.html?gmcCountry=US&currency=USD&createTmp=1&utm_source=googleshopping&utm_medium=cpc_bgs&utm_content=frank&utm_campaign=pla-elc2-us-pc-0720&ad_id=368005069105&gclid=EAIaIQobChMI2pfb0Zj54wIVmeDICh2egQ_iEAQYAyABEgLVgvD_BwE&cur_warehouse=CN

Or this one: https://www.digikey.com/reference-designs/en/ac-dc-and-dc-dc-conversion/dc-dc-smps-single-output/1592?utm_adgroup=Converters&gclid=EAIaIQobChMIvM6-9pf54wIVxMDICh36XAxlEAAYASAAEgJsF_D_BwE

Or this one: https://www.adafruit.com/product/2190

In a word, no. You need to provide an Arduino Micro with either > 6.5V unregulated into the barrel connector, or a regulated 5V supply directly to the 5V line.

You could use a buck/boost DC/DC converter to boost your 3.7V battery output to 5V and feed that directly into the 5V line on the Arduino (or better, feed it into the USB connector, since that will use the input switching on the Arduino.)

This one, for example:

https://www.banggood.com/DC-DC-3V-15V-to-5V-Fixed-Output-Automatic-Buck-Boost-Step-Up-Step-Down-Power-Supply-Module-p-1355829.html?gmcCountry=US&currency=USD&createTmp=1&utm_source=googleshopping&utm_medium=cpc_bgs&utm_content=frank&utm_campaign=pla-elc2-us-pc-0720&ad_id=368005069105&gclid=EAIaIQobChMI2pfb0Zj54wIVmeDICh2egQ_iEAQYAyABEgLVgvD_BwE&cur_warehouse=CN

Or this one: https://www.digikey.com/reference-designs/en/ac-dc-and-dc-dc-conversion/dc-dc-smps-single-output/1592?utm_adgroup=Converters&gclid=EAIaIQobChMIvM6-9pf54wIVxMDICh36XAxlEAAYASAAEgJsF_D_BwE

Or this one: https://www.adafruit.com/product/2190