I am glad to see this question here because, 7 years ago, I joined the electronics forum of ResearchGate with the same question, "As the virtual ground is a kind of ground, can we connect it to the real ground?" ... and I received 38 answers (a good achievement for a newcomer). A few years later, after asking more 123 questions, I moved to SE EE… and now I am answering the same question here. But I realize that after all these years, I should say something better and more intriguing.
Although the virtual ground evokes a sense of mystery, I will try to show that it is a simple concept. The paradox is that I have to use a lot of words for this purpose... but I do not see other way to convince you...
The main task of the op-amp in negative feedback circuits is to maintain (almost) zero voltage between its two inputs. This creates the illusion of a short circuit between the inputs even though no current flows between them. Since this is not a real short circuit, we call it "virtual short circuit" or simply "virtual short" (in circuit theory, it is named "nullator"). What is important to understand for beginners is that the short circuit is not internal between the inputs but external (by the network of two elements in series - R2 and VOUT). The op-amp can maintain the equality between the two input voltages in several ways:
1. Changing the voltage of the inverting input (an example are inverting circuits with negative feedback)
2. Changing the voltage of the non-inverting input (non-inverting circuits with negative feedback)
3. Simultaneously changing both voltages (negative impedance converters - NIC, having both negative and positive feedback)
While in the last two cases above the "virtual short" is floating, in the first case its non-inverting end is grounded and we say that the inverting end is a "virtual ground". Or the virtual ground is a node that is virtual short connected with the real ground. Thus the virtual ground is just a copy of the real ground... a node whose voltage copies (follows) the voltage of the real ground… a clone of the real ground. So it is implemented by the output of a voltage follower whose input is connected to the real ground. But what is this follower in the OP's circuit?
My answer is simple and maybe surprising: The whole circuit of four resistors and op-amp is a follower… but a "disturbed follower". I will explain what I mean by showing the evolution of the circuit from an undisturbed to a disturbed follower.
1. Undisturbed follower. If we connect the op-amp inverting input to its output, and the non-inverting input to ground, we obtain the classic op-amp voltage follower - Fig. 1.
Fig. 1. Classic op-amp voltage follower with zero input voltage (zero voltage stabilizer)
The op-amp will do whatever to keep (almost) zero voltage between its inputs (the H&H's "golden rule"). So the voltage of the inverting input is a copy of the ground voltage (almost zero)... and the circuit acts as a zero voltage stabilizer with negative feedback. This node serves as another but artificial ("virtual") ground.
2. VIN+R disturbed follower. Now we decide to see if this stabilizer is reliable… and, for this purpose, we intend to "provoke" it with another voltage source. But let's not be so brutal and connect it through a resistor to this node - Fig. 2.
Fig. 2. Disturbed but not reacting op-amp follower with zero input voltage (disturbed zero voltage stabilizer)
The (input) voltage source "pulls down" the virtual ground... but the op-amp, behaving as a perfect voltage source, does not allow it to *move". So, the op-amp does not react noticeably to this additive disturbance. We have to do something else...
3. VIN+R1+R2 disturbed follower. The problem is that the two sources try to set the voltage of the same node but they are not under the same conditions - VOA does it directly while VIN through a resistor. Then let's make it difficult for VOA by connecting another resistor R2 in series - Fig. 3.
Fig. 3. Once-disturbed op-amp follower (inverting amplifier)
The resistors R1 and R2 constitute a voltage divider acting as a multiplicative disturbance (beta1) for the op-amp. To compensate it, the op-amp increases R2/R1 times its output voltage… thus acting as an inverting amplifier (if we take this voltage as an output VOUT).
Very interesting - we managed to explain in a logical way the need for both resistors R1 and R2 in an inverting amplifier! But note the voltage follower is still here and it continues keeping steady zero voltage at the virtual ground node.
4. VIN+R1+R2+R3+R4 disturbed follower. If we want a higher gain, the ratio R2/R1 becomes too high... and we decide to put another multiplicative disturbance beta2 (voltage divider R3-R4) in the feedback loop - Fig. 4.
Fig. 4. Twice-disturbed op-amp follower (inverting amplifier with T feedback network)
The op-amp reacts to this intervention by increasing more (1 + R4/R3) its output voltage. So, this "twice-disturbed follower" acts as an amplifier with a higher gain. Note that the previous voltages remain and there are in total three circuit outputs - the undisturbed VOUT1, once-disturbed VOUT2 and twice-disturbed VOUT3... but here we are interested only in the first.
The conclusion is that when disturbing more and more the op-amp, it overcomes any disturbances (here, one additive and two multiplicative) in the name of maintaining zero voltage at its inverting input (of course, its output voltage must not reach the supply rails). In its quest to achieve this, the follower becomes an amplifier.
It is so interesting conclusion: An amplifier with negative feedback is actually a disturbed zero voltage follower! And this is the recipe how to convert a follower into an amplifier - by disturbing it.
We can see this technique everywhere around us... and even here, in SE EE:) Let's consider this intriguing psychological phenomenon.
As you know, there are people who regularly downvote our answers; thus they regularly "disturb" us. But since we behave (like the op-amp above) as a negative feedback system with a goal, we overcome these "disturbances" and write even better answers.
This was my incredible story about the virtual ground… particularly, in this circuit solution aka "inverting amplifier with T feedback network". I hope that now you will better understand why you should not connect the virtual ground to real ground...