Yes, as long as you stay under the maximum output current rating.
Eight Power DMOS-Transistor Outputs of 150-mA Continuous Current.
Relays are to have kickback diodes across the coils.
Unnecessary with the TPIC6x595.
The ULN2x03 contains such diodes but you need to connect the common to the supply .
Unnecessary with the TPIC6x595.
That diagram is a logic diagram, not a circuit diagram. Read the datasheet. 
That schematic is the equivalent circuit for the TPIC6x595 chip.
The response was such as if the questioner was connecting the TPICs to simple relays and the Arduino.
Sorry, it is not. It is a logic diagram. You cherry-picked it because it seemed to suit your preconceived notion.
You didn't actually read the datasheet, did you?
Page 1:
3 Description
The device contains a built-in voltage clamp on the outputs for inductive transient protection. Power driver applications include relays, solenoids, and other medium current or high-voltage loads.
Page 11, immediately before the diagram you quoted:
8.1 Overview
The TPIC6B595 device is a monolithic, high-voltage, medium-current power 8-bit shift register designed for use in systems that require relatively high load power. The device contains a built-in voltage clamp on the outputs for inductive transient protection, so it can also drive relays, solenoids, and other medium-current or high-voltage loads.
And following the very diagram you quoted:
The output clamp voltage is 50V.
This is still too high and causes problems in near by traces.
A diode across inductive loads will prevent any noise problems from being picked up.
In practice placing a high voltage clamping diode across BJTs and MOSFETs is not recommended in industrial situations.
We seem to have to go through this every year.
One last time.
Here we are keying a 5-volt relay with the output of a TPIC6B595.
We can see the kickback spike of about 54 volts.
This spike has a duration of about 85us.
What is more important is the Rise Time is about 450ns.
I am sure we can all agree that a 55-volt spike with a rise time of 450ns will have a huge number of high frequency components.
The problem is this high frequency noise can and does cause problems with near by traces on the PCB.
Cabling noise going to the relays can cause havoc with circuitry.
A clamping diode across an active device for kickback purposes is not recommended.
A simple 5 cent diode cross the relay coil prevents this phenomenon from happening.
53 V actually. 
Nonsense. Only four and a bit times the usual relay/ solenoid voltage. 12 V in this case.
Also nonsense. The rise time will be the same. it is a function of the performance of the FET. If you choose to run sensitive tracks near the relay outputs, more fool you!
Not recommended - by you?
50 V Zener plus 3 V on the FET gate.
So four times that duration if you put a diode across the inductor (or ten times that for a 5 V relay). That is "slugging" the relay.
So if you put a diode directly across the relay, the rise time is proportionately shorter. 100 ns for 12 V, 45 ns for 5 V by definition.
Same number or proportion for all the alternatives. Only the voltage changes.
Point of design.
An awful lot worse if you put a diode across the relay itself!
By LarryD. Indeed. 
Just - no. It does not.
Actually putting it directly across the relay coil can make it worse.
Thanks for LarryD and Paul_B for the detailed discussion and practical example.
I use TPIC6B595DWRG4 and observed very sensitive with noise. I also use kickback diodes with relay and 0.1uf capacitor. With this arrangement it was getting some noise. I didn't understand the reason but I solve this issue by pull up all free pins on arduino pro mini.
My relays are getting hot after few minutes of running is there is some issue in design. Voltage on relay coil is 11.5V.
I use this relay 133-1A-12DS
Hope you are going to feel much better soon.
How hot ?
12v / 320 Ω = 37mA
Only about 0.44W 
BTW, how much current flows thru the contacts.
Without load
At ≈1/2 Watt, I would expect warm, not hot.
Please confirm you have decoupling on the power pin ?
yes 0.1uf near to the ic and 0.1uf+10uf on voltage regulator
Both leads, i.e. near to IC ground and near to IC Vdd also? Thick traces? What is the component designation? I don't see it there. Is it C1? C1 appears to have a meandering trace on one side...
Ideally it should be right on pin 2, with a wide trace. You've got it the other end of a long thin signal trace several cm away though. At the frequencies involved every cm of trace is a significant impedance due to the stray inductance. Its best to aim for each power pin to have its fast decoupling within a few mm, and all ground pins to directly connect to groundplane. Fast decopling has to work at nanosecond timescales.
I suspect your layout is almost certainly going to work here, but chips are only getting faster and its good to get into good habits early. Some modern chips require literally dozens of decoupling caps as close as possible to the pins (often BGA), so they have to be on the back of the pcb in order to be close enough.