I'm afraid that WhatsApp for Android does not currently support to be called from a web browser.
I had the same requirement for my current project, and since I couldn't find any proper information I ended up downloading the APK file.
In Android, if an application wants to be called from a web browser, it needs to define an Activity with the category android.intent.category.BROWSABLE.
You can find more information about this here: https://developers.google.com/chrome/mobile/docs/intents
If you take a look to the WhatsApp AndroidManifest.xml file, the only Activiy with category BROWSABLE is this one:
<activity android:name="com.whatsapp.Conversation" android:configChanges="keyboard|keyboardHidden|orientation|screenLayout|uiMode|screenSize|smallestScreenSize" android:windowSoftInputMode="stateUnchanged">
<intent-filter>
<action android:name="android.intent.action.SENDTO" />
<category android:name="android.intent.category.DEFAULT" />
<category android:name="android.intent.category.BROWSABLE" />
<data android:scheme="sms" />
<data android:scheme="smsto" />
</intent-filter>
</activity>
I've been playing with it for a while, and I couldn't make it to work. The most I got was to open the WhatsApp application from Chrome, but I couldn't figure out a way to set the message content and recipient.
Since it is not documented by the WhatsApp team, I think this is still work in progress. It looks like in the future WhatsApp will handle SMS too.
The only way to get more information is by reaching the WhatsApp dev team, what I tried, but I'm still waiting for a response.
Regards!