When published application or desktop is launched through browser, the ICA file automatic launch is handled by the protocol handle of the browser itself. The following components plays its own role to achieve the automatic launch:
Storefront server
If on-prem Storefront server is used in the deployment,  Storefront Management Console > you store > Manage Receiver for Web site > Advanced Settings > Enable protocol handle must be ticked to use protocol handle of the browser.

Citrix Workspace

Install or Update Citrix Workspace App
The most common cause of .ica file issues is a missing or outdated Citrix client.
Check for Installation: Ensure the Citrix Workspace App is installed on your computer.
Update to Latest Version: If it's installed, update it. Many recent browser/OS issues have already been fixed in the latest releases.
Download Link: Download Citrix Workspace App Here 
Alternative: If you cannot install the client, use the Citrix Workspace App for HTML5 option on your Storefront page (if available) to launch the session directly in the web browser.

 
Browser
1.When access the store, for the first time or when the browser is launched in InPrivate /Incognito/ window, tick the "Always allow  [Your Store URL] to open links of this type in the associate app" then click "Open"

2.Each browser has its own policy/configuration to associate the default application for a file type. 
The protocol handler called "receiver" needs to be register to the browser protocol handler accordingly.
Policy settings are available for managed-devices on Chrome and MS edge, the steps below are for your reference only. Please consult each browser vendor for detailed configuration instructions.

Chrome: 
Download ADMX template for Chrome browser
2. Copy .ADMX files to C:\Windows\PolicyDefinitions folder and the relevant language specific ADML files to C:\Windows\PolicyDefinitions\xx-xx folder. 
For example, if you are using en-US language, copy the .adml files under "en-US" folder to C:\Windows\PolicyDefinitions\en-US folder.
3. Run group policy > Computer Configuration >Administrative Templates > Google >Google Chrome - Default Settings (users can override)  >Content settings > Register protocol handlers 

[
  {
    "default": true,
    "protocol": "receiver",
    "url": "your store URL"
  }
]
Reference: https://developer.chrome.com/docs/web-platform/best-practices/url-protocol-handler

MS Edge:
Download the MS Edge policy from https://www.microsoft.com/en-us/edge/business/download?cs=3457492030&form=MA13FJ
Copy .ADMX files to C:\Windows\PolicyDefinitions folder and the relevant language specific ADML files to C:\Windows\PolicyDefinitions\xx-xx folder. 
group policy > Computer Configuration >Administrative Templates > Microsoft Edge - Default Settings (users can override)/Content settings RegisteredProtocolHandlers
[{"default": true, "protocol": "receiver", "url": "your store URL"}]
Reference: https://learn.microsoft.com/en-us/deployedge/microsoft-edge-browser-pol
icies/registeredprotocolhandlers

Firefox
Download the MS Edge policy from  https://github.com/mozilla/policy-templates/releases

Copy .ADMX files to C:\Windows\PolicyDefinitions folder and the relevant language specific ADML files to C:\Windows\PolicyDefinitions\xx-xx folder. 
group policy > Computer Configuration >Administrative Templates > Firefrox > Extensions > Extension Management (JSON on one line)
{
  "protocol_handlers": [
    {
      "protocol": "receiver",
      "url": "/lookup?type=%s"
    },

Advanced Troubleshooting (General / Legacy Browser Issues)

If the above steps don't fix the issue, especially on older systems or where strict security policies are in place, try these steps.

General Security Checks

• Remove Adware: Temporarily remove or disable any third-party browser-based Adware or aggressive browser extensions that could interfere with the file download and launch process. Restart your browser.

• Check Antivirus/Firewall: Ensure your local antivirus or firewall is not blocking the Citrix Workspace App or the downloaded .ica file.

Internet Explorer (IE) Security Settings

(⚠️ Note: These are for older, less secure IE deployments and are generally not recommended for modern browsers.)

For Administrators: Understanding the Default Protocol

When the Citrix Workspace App is installed correctly, the operating system registers a custom protocol handler called receiver:// to manage .ica files. The browser passes the launch request to this protocol, which is handled by WebHelper.exe on Windows:

HKEY_CLASSES_ROOT\receiver\shell\open\command

(Default): "C:\Program Files (x86)\Citrix\ICA Client\WebHelper.exe" "%1"

The purpose of the browser fixes above is to get the browser to correctly detect and use this registered receiver:// protocol instead of just downloading the file.