In the United States, shaking hands with your left hand is generally considered unusual and may be perceived as disrespectful or awkward. The right hand is the customary hand for handshakes in American culture and many other cultures around the world.
There are historical reasons for this preference. In the past, most people were right-handed, and the right hand was used for handshakes as a sign of trust and goodwill. Using the left hand for greetings or handshakes was often considered inappropriate because historically the left hand was associated with tasks considered unclean, like personal hygiene. Today, while the association with cleanliness has largely disappeared, the tradition of using the right hand for handshakes remains.
If you accidentally offer your left hand for a handshake in America, most people will likely understand that it was not intentional and may not be offended. However, it's a good idea to be mindful of cultural norms when interacting with people from different backgrounds.
In some cultures, the left hand is still considered unclean, and using it for greetings or handshakes can be genuinely disrespectful. So, if you are in a diverse cultural setting, it's essential to be aware of the customs and norms of the specific group you are interacting with to show respect and avoid any unintended offense. When in doubt, it's always best to follow the local customs and use your right hand for handshakes in most Western cultures, including the United States.