Yes use of calloc()-allocated memory will suffer a performance degradation due to the Meltdown and Spectre patches. 
In fact, calloc() isn't special here: malloc(), new and more generally all allocated memory will probably suffer approximately the same performance impact. Both calloc() and malloc() are ultimately backed by pages returned by the OS (although the allocator will re-use them after they are freed). The only real difference being that a smart allocator, when it goes down the path of using new pages from the OS (rather than re-using a previously freed allocation) in the case of calloc it can omit the zeroing because the OS-provided pages are guaranteed to be zero. Other than that the allocator behavior is largely the same and the OS-level zeroing behavior is the same (there is usually no option to ask the OS for non-zero pages).
So the performance impact applies more broadly than you thought, but the performance impact is likely smaller than you suggest, since a page fault is already doing a lot of work anyways, so you aren't talking an order of magnitude degradation or anything. See Peter's answer on the reasons the performance impact is likely to be limited. I wrote this answer mostly because the answer to your headline question is still yes as there is some impact.
To estimate the impact on a malloc heavy workflow, I tried running some allocation and page-fault heavy test on a current kernel (4.13.0-39-generic) with the Spectre and Meltdown mitigations, as well as on an older kernel prior to these mitigations.
The test code is very simple:
#include <stdlib.h>
#include <stdio.h>
#define SIZE        (40 * 1024 * 1024)
#define PG_SIZE     4096
int main() {
    char *mem = malloc(SIZE);
    for (volatile char *p = mem; p < mem + SIZE; p += PG_SIZE) {
        *p = 'z';
    }
    printf("pages touched: %d\npoitner value : %p\n", SIZE / PG_SIZE, mem);
}
The results on the newer kernel were about ~3700 cycles per page fault, and on the older kernel without mitigations around ~3300 cycles. The overall regression (presumably) due to the mitigations was about 14%. Note that this in on Skylake hardware (i7-6700HQ) where some of the Spectre mitigations are somewhat cheaper, and the kernel supports PCID which makes the KPTI Meltdown mitigations cheaper. The results might be worse on different hardware.
Oddly, the results on the new kernel with Spectre and Meltdown mitigations disabled at boot (using spectre_v2=off nopti) were much worse than either the new kernel default or the old kernel, coming in at about 5050 cycles per page fault, something like a 35% regression over the same kernel with the mitigations enabled. So something is going really wrong, performance-wise when the mitigations are disabled. 
Full Results
Here is the full perf stat output for the two runs.
Old Kernel (4.10.0-42)
pages touched: 10240
poitner value : 0x7f7d2561e010
 Performance counter stats for './pagefaults':
         12.980048      task-clock (msec)         #    0.976 CPUs utilized          
                 0      context-switches          #    0.000 K/sec                  
                 0      cpu-migrations            #    0.000 K/sec                  
            10,286      page-faults               #    0.792 M/sec                  
        33,662,397      cycles                    #    2.593 GHz                    
        27,230,864      instructions              #    0.81  insn per cycle         
         4,535,443      branches                  #  349.417 M/sec                  
            11,760      branch-misses             #    0.26% of all branches        
0.013293417 seconds time elapsed
New Kernel (4.13.0-39)
pages touched: 10240
poitner value : 0x7f306ad69010
 Performance counter stats for './pagefaults':
         14.789615      task-clock (msec)         #    0.966 CPUs utilized          
                 8      context-switches          #    0.541 K/sec                  
                 0      cpu-migrations            #    0.000 K/sec                  
            10,288      page-faults               #    0.696 M/sec                  
        38,318,595      cycles                    #    2.591 GHz                    
        28,796,523      instructions              #    0.75  insn per cycle         
         4,693,944      branches                  #  317.381 M/sec                  
            26,853      branch-misses             #    0.57% of all branches        
       0.015312764 seconds time elapsed
New Kernel (4.13.0.-39) spectre_v2=off nopti
pages touched: 10240
poitner value : 0x7ff079ede010
 Performance counter stats for './pagefaults':
         16.690621      task-clock (msec)         #    0.982 CPUs utilized          
                 0      context-switches          #    0.000 K/sec                  
                 0      cpu-migrations            #    0.000 K/sec                  
            10,286      page-faults               #    0.616 M/sec                  
        51,964,080      cycles                    #    3.113 GHz                    
        28,602,441      instructions              #    0.55  insn per cycle         
         4,699,608      branches                  #  281.572 M/sec                  
            25,064      branch-misses             #    0.53% of all branches        
       0.017001581 seconds time elapsed