On a public account, yes. A story's viewer list is made of logged-in Instagram accounts that opened it. Watch the story without logging in to Instagram, and there is no account of yours for that list to show.
Two things stay out of reach whatever you use: the stories of a private account, and stories shared only with Close Friends.

How Instagram decides who viewed a story
When you open a story in the Instagram app, two separate things happen. The app loads the picture or the video, and then it tells Instagram that your account has seen it. The second step is what puts your name in the list the owner reads.
A story on a public account can be loaded without that second step. Nobody is logged in, so nothing is marked as seen by you. That is all an honest story viewer does, and it is why none of them should ever ask for your Instagram password.
Watch Instagram stories without an account
You don't need an Instagram account at all. Instagram's own website asks you to sign in before it plays a story, so you go through a viewer instead: a website or an app that loads the public story for you. No account of yours, nothing of yours in the list.
What does not work as well as people say
A second Instagram account
It keeps your main name out of the list, and puts the second account in it instead. If the owner knows, or guesses, who is behind it, nothing was hidden.
Airplane mode
The old trick: let the stories load, switch to airplane mode, watch, close the app. Nobody outside Instagram can confirm what the app sends once it reconnects, and you have no way to check from your side. It is a gamble, not a method.
Anything that asks for your login
A viewer that wants your Instagram username and password is the opposite of what you came for: it would watch as you. It also breaks Instagram's terms and puts your own account at risk.
What story viewer websites do
You type a username, a server loads the account's current stories, and you watch them in the browser. On a public account that works, on any phone, and many of these sites are free.
What a website cannot do is tell you when there is something new. Stories last 24 hours, and most people do not post every day, so you end up checking a page that is empty most of the time, and missing the story that went up and came down while you were not looking.
An Instagram story viewer app for Android
FollowBell is an Android app that watches public Instagram accounts. Stories are part of that.
- A ring when there is a story. The account's photo gets a ring while a story is up, the way Instagram shows it.
- A notification when a new one is posted. For every account you watch, on by default, and you can turn it off account by account.
- A full-screen player. Tap to skip, hold to pause, sound on videos, and how long ago each story was posted.
- No login, at any point. FollowBell never asks for an Instagram account, so the player says what is true: viewed without appearing in the viewer list.
There is no Open in Instagram button in the player, on purpose. Opening the story there, with your own account, would put you in the list you just stayed out of.
When there is no story, FollowBell says so plainly and shows the account's highlights instead. How highlights work.
What you can do: watch the current stories of any public account, without an Instagram login and without your name in its viewer list.
What no app can do, whatever its ads say: show a private account's stories, show Close Friends stories, or bring back a story that expired and was never saved to a highlight.
What stays out of reach, for every tool
- Private accounts. Their stories are for the people they accepted. Any site claiming otherwise is making it up.
- Close Friends stories. Shown only to the people on that list, even on a public account.
- Expired stories. After 24 hours a story is gone from Instagram, unless it was saved to a highlight. FollowBell does not keep copies, and does not let you download or share a story.
- Who viewed someone's profile. Instagram publishes it to nobody. A tool that promises it is inventing the answer.