I do not have any other computer available at the moment. I could try installing it in a friend's computer. But not before tomorrow.

Since your other card worked, no need to try the other slots not intended for GPU
 
Yes, double check install, using a good light helps see what is going on
 
Could be a PSU issue or the new card could have an issue & sometimes newer GPU are not compatible with older MB - but I would think it would still try to POST.  That is why I think it is a power issue - connection

FranKDressler
I do not have any other computer available at the moment. I could try installing it in a friend's computer. But not before tomorrow.

Let us know how it works in another pc.

Your PSU comes with 4 Each PCI-E 6+2 pin (GPU) Cables you should NOT be using the Adaptor that came with your Graphics Card.
You can only use 1 x PCIe 3.0/2.0 x16 (blue) for your Graphics Card, the Slot next to the CPU Socket.
 Remove your CMOS Battery AND Power Cable for 10 Minutes.
Disable (Initiate iGPU) LucldLogix Virtu MVP in your Bios. *If you installed this with your 660 GPU put back in the 660 card and uninstall the software and then disable it in the Bios.
Page 3-18 Global
In your Bios set the Primary Display to PEG
In your Bios Disable (Initiate iGPU) LucldLogix
In your Bios Set the PCIEx16 1 Link Speed to GEN3
 
1 x PCIe 3.0/2.0 x16 (blue)
1 x PCIe 2.0 x16 (x4 mode, black) *¹
2 x PCIe 2.0 x1 
3 x PCI
*¹ The PCIe x 16 slot (black x4 mode) is compatible with PCIe x1 and x4 devices.
What other PCIe Devices do you have Installed?
What other PCI Devices do you have Installed?
 
 https://www.asus.com/Motherboards/P8H77V/
You may need to Open a Support Ticket on ASUS's website.
 
Are the two below install?
Management Engine Interface V11.0.0.1157 for Windows 10 32bit & 64bit---(WHQL). Global
Intel Chipset Driver V10.1.1.7 for Windows Win10 32bit & 64bit. Global
 
P8H77-V User's Manual (English) Global
I Hate PDF files that I cannot copy the Text.

I have tried what you suggested bcavnaugh, but without success.

The manual and the actual UEFI differ:
    System Agent\Graphics Configuration\
        Primary Display:
            Options:
                Manual: [Auto], [iGFX], [PEG], [PCI]
                Actual: ..., [PCIe],...
            Set: [PCIe]
            
        Inititate iGPU:
            Options:
                Manual: [Enabled], [Disabled]
                Actual: Does not exist at all
            
    System Agent\NB PCIe Configuration\
        PCIEX16_1 Link Speede:
            Options:
                Manual: [Auto], [Gen1], [Gen2], [Gen3]
                Actual: Same
            Set: [Gen3]
               
I have set the options you suggested with my old GTX 660 installed. After setting, saving and rebooting, I directly went back into the BIOS to verify that the settings were still as I set them. As this was the case, I shut the system down, installed the 1070, but the system still didn't turn on. So I removed the card and booted into Windows using the iGPU. The system automatically enables the iGPU ([Auto]) if the primary display is set to PCIx and there is no PCIx graphics card installed. The system still works well with the GTX 660.

Management Engine Interface and Intel Chipset Driver are not installed. But the system does not even reach the state at which it matters, I guess. Because nothing happens when I press the power button while the 1070 is installed.

I have also tried resetting the CMOS by removing the battery. Didn't help.

I have created a support ticket yesterday and the support's response to it allows me to start an RMA process. As the friend in whose computer I wanted to test the card is not available, I am going to start the RMA process. His computer would also not be much different from mine because he bought it at about the same time as I did mine and with similar components.
 
Trying to figure out if the board and card are compatible, I was searching for a database that contains working hardware combinations. Then I got the idea that there are benchmark websites with user reported benchmarks. So I googled for "p8h77-v gtx 1070 benchmark" and have found a list of benchmarks on the Internet in which the 1070 GTX was benchmarked with the board, P8H77-V, that I have (UserBenchmark: 1857424, P8H77-V + GTX 1070). So I assume the combination should work, unless the user has faked his report.
 
To summarize the state:
1. Benchmark (UserBenchmark: 1857424) shows the board and card are compatible. [Edit: UserBenchmark address: userbenchmark . com/UserRun/1857424]
2. PC + iGPU, no extra graphics card: works
3. PC + GTX 660 Ti: works
4. PC + GTX 660 Ti without PCIe-Power connected: Display shows message like "Please connect the PCIe-Power cables"
5. PC + GTX 1070: computer does not start
6. PC + GTX 1070 without power connected: computer starts but without message like "Please connect the PCIe-Power cables". Graphics card outputs no image and iGPU works. Windows device manager shows no GTX 1070.
7. BIOS settings: Setting primary display to PEG/PCIe, PCIx-Port set to Gen 3.0: Does not make it work.
8. Switching between Master/Slave on GTX 1070: Does not make it work.
 
So I conclude:
1. Card and board are compatible because of #1
2. PCI port works because of #3.
3. PSU works because of #3+#4.
4. Cables probably also work because of #3+#4. Only probably: The GTX 1070 FTW uses 2 x 8 Pins, the GTX 660 Ti that I have 2 x 6. But I have tried 2 different 2 x (6 + 2) cables that come with my PSU.
 
And hence I believe the card that I bought is broken.
 
Thank you all for your efforts to help me. I am going to report the progress of the RMA process in case that might be interesting, and if the replacement card, that I assume to get, works.

Quad5Ny
I've been wrong before though. :P

You Have

One Card is UEFI based and the Other Card is Not UEFI based.
Not all older motherboards support UEFI Graphics Cards, But this MB supports UEFI
The GTX 1070 is a UEFI Graphics Card.
The GTX 660 is Not a UEFI Graphics Card.
I have read here on the Forum their is a Bois update for the GTX 660 to run UEFI.
 

The manual of the MB and reality do not seem to know each other. The BIOS offers a screenshot function that I used to create the following screenshots.
 
The boot tab contains an entry for CSM as shown in the following images.
Boot: Part 1

Boot: Part 2

 
This is the Boot\CSM menu
   
and the chosen settings

And in the Boot\Secure Boot entry shows

I set the PCIE to Gen 3 as shown in this image

And primary display to PCIE as mentioned earlier

but it reset itself to Auto when I removed the card when it did not work.
Finally the main screen of the BIOS with the version information

 
But as I said, I believe all this does not matter because the system does not even power on noticably when the card is installed.

FranKDressler
The manual of the MB and reality do not seem to know each other. The BIOS offers a screenshot function that I used to create the following screenshots.
 
The boot tab contains an entry for CSM as shown in the following images.
Boot: Part 1

Boot: Part 2

 
This is the Boot\CSM menu
   
and the chosen settings

And in the Boot\Secure Boot entry shows

I set the PCIE to Gen 3 as shown in this image

And primary display to PCIE as mentioned earlier

but it reset itself to Auto when I removed the card when it did not work.
Finally the main screen of the BIOS with the version information

 
But as I said, I believe all this does not matter because the system does not even power on noticably when the card is installed.

It's not the bios. How old is your PSU.

Messing with bios setting isn't going to solve the OP's problem as the computer doesn't even start when the card is inserted into the motherboard.