Seems to me that it does indeed support push:
Mail, Calendar, & Contacts: Set Up Your Apple Device for Google Sync
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To set up Google Sync on your iPhone, iPad or iPod Touch device, please follow the steps below.
Requirements and Upload
1. Google Sync is only supported on iPhone OS versions 3.0 and above. You can check your current version by going to
Settings >
General >
About >
Version. To upgrade, follow the instructions at
Apple - iOS 4.2 Software Update. Please upgrade to iPhone OS 3.0 before setting up Google Sync.
2. Perform a sync with iTunes to ensure that Contacts and Calendars from your iPhone are backed up to your computer. Learn more about backing up your
Contacts and Calendars.
3. If your business, school, or organization uses Google Apps, your administrator will first need to
enable Google Sync before you can take advantage of this feature. Note that if you enable Google Sync, your administrator may also exercise additional remote management capabilities (such as remote wipe or password requirements).
Click to learn more or contact your administrator.
Getting Started
1. Open the
Settings application on your device's home screen.
2. Open
Mail, Contacts, Calendars.
3. Tap
Add Account....
4. Select
Microsoft Exchange. OS 4.0+ now allows multiple Exchange accounts. However, if you're on a device that doesn't let you add a second account, you could also
use CalDAV to sync Google Calendar and
IMAP to sync Gmail.
Enter Account Info
5. In the
Email field, enter your full Google Account email address. If you use an @googlemail.com address, you may see an "Unable to verify certificate" warning when you proceed to the next step.
6. Leave the
Domain field blank.
7. Enter your full Google Account email address as the
Username.
8. Enter your Google Account password as the
Password.
9. Tap
Next at the top of your screen.
9a. Choose
Cancel if the
Unable to Verify Certificate dialog appears.
10. When the new
Server field appears, enter
m.google.com.
11. Press
Next at the top of your screen again.
12. Select the Google services (Mail, Calendar, and Contacts) you want to sync.
13. Unless you want to delete all the existing Contacts and Calendars on your phone, select the
Keep on my iPhone option when prompted. This will also allow you to keep syncing with your computer via iTunes.
If you want to sync only the
My Contacts group, you must choose to
Delete Existing Contacts during the Google Sync install when prompted. If you choose to keep existing contacts, it will sync the contents of the "All Contacts" group instead. If there are no contacts on your phone, the latter will happen -- the contents of your
All Contacts group will be synced.
You've set up Google Sync! Synchronization will begin automatically if you have Push enabled on your device. You can also open the Mail, Calendar or Contacts app and wait a few seconds to start a sync.
Next
choose which calendars to sync, if you'd like to enable multiple calendars.
