Can i put a sim card in my apple watch 6.1.3 to use internet even without carrying the phone with me?

None of the Apple watches have a slot.

You need to have a paired iPhone

Even if you don't have your phone with you it will need to be turned on

Have read https://support.apple.com/en-ca/HT207578

As correctly pointed out by Randy-12 and fieryjack, that will not be possible.

Bear in mind that for each model of the Apple Watch, there are two versions - GPS and GPS + Cellular

GPS - has only Wifi and your iPhone must be close by for some iPhone cell/data functions - receiving/making a call, for example.

GPS + Cellular - has the possibility of independant cellular/data functions via eSIM and does not need to have the iPhone close by.

In either version, there is NO possibility of using a regular/physical nano SIM card.

Matricky wrote:

Thank you. But does my watch (6.1.3), after it has been paired with my phone, works independently? Ie can I use internet, pay with my saved cards, etc without carrying my phone? Thx!

You have not mentioned which series Apple Watch you are using.

I think, it must be Series 3 (or newer).

When you say 6.1.3, I think you are referring to watchOS which depending on which Apple Watch you are using, may be upgradable. Update your Apple Watch - Apple Support

Your Apple Watch has WiFi ... so you can connect to the internet using WiFi, as long as the WiFi network is compatible with Apple Watch. This connection is usually done via your linked iPhone, but it can also be done without the iPhone. Once you have WiFi on your Apple Watch, certain online (internet) tasks can be done!

Connect your Apple Watch to Wi-Fi - Apple Support

About Bluetooth, Wi-Fi, and cellular on your Apple Watch - Apple Support

Using just your Apple Watch even when you do not have your linked iPhone around, you have the possibility of using Apple Pay in some instances. Once you set up your cards using the Apple Watch app for iPhone, you don’t need the iPhone around to make a purchase, since the Apple Watch has its own Secure Element to store your device’s ID number.

Apple Pay on Apple Watch - Apple Support

So, yes, you can use your Apple Watch to pay at stores which have the Apple Reader!

But, let's say you want to do Person to person payments via Apple Pay by sending a message, that will require your Apple Watch to have a connection to the internet... and if you do not have one (either WiFi or Cellular), then you cannot send the message and hence make the payment!

Some additional information: What can your Apple Watch do without your iPhone? - macworld.com

No,a cellular watch has a built in Esim that uses your phones number,there is no physical sim card and if it's a GPS only watch then you have to carry your phone with you.

It depends what form of payments.